Installing and Loading Required Libraries

# # Required Packages
packages = c('quantmod','car','forecast','tseries','FinTS', 'rugarch','utf8','ggplot2')
# 
# # Install all Packages with Dependencies
# install.packages(packages, dependencies = TRUE)
# 
# # Load all Packages
lapply(packages, require, character.only = TRUE)
[[1]]
[1] TRUE

[[2]]
[1] TRUE

[[3]]
[1] TRUE

[[4]]
[1] TRUE

[[5]]
[1] TRUE

[[6]]
[1] TRUE

[[7]]
[1] TRUE

[[8]]
[1] TRUE
library(writexl)
library(tseries)
library(TSstudio)
library(fBasics)
library(rcompanion)
library(forecast)
library(lmtest)
library(tsDyn)
library(vars)
library(PerformanceAnalytics)
library(vrtest)
library(pracma)
library(rugarch)
library(FinTS)
library(e1071)
library(readxl)
# Fetching Data
symbols <- c("^BSESN","^GSPC","^N225","^HSI","^N100")
getSymbols(Symbols = symbols,
           src = 'yahoo',
           from = as.Date('2018-01-01'),
           to = as.Date('2023-12-31'),
           periodicity = 'daily')
Warning: ^BSESN contains missing values. Some functions will not work if objects contain missing values in the middle of the series. Consider using na.omit(), na.approx(), na.fill(), etc to remove or replace them.
Warning: ^N225 contains missing values. Some functions will not work if objects contain missing values in the middle of the series. Consider using na.omit(), na.approx(), na.fill(), etc to remove or replace them.
Warning: ^N100 contains missing values. Some functions will not work if objects contain missing values in the middle of the series. Consider using na.omit(), na.approx(), na.fill(), etc to remove or replace them.
[1] "BSESN" "GSPC"  "N225"  "HSI"   "N100" 

The names of the stock exchanges selected

1.  **\^BSESN** - SENSEX (BSE Sensex) of the **Bombay Stock Exchange (BSE)-**

The Bombay Stock Exchange (BSE), located in Mumbai, India, is one of the oldest stock exchanges in Asia, founded in 1875. The BSE Sensex, or SENSEX, is its benchmark index, representing 30 of the largest and most actively traded stocks on the exchange, reflecting the overall performance of the Indian stock market.

2.  **\^GSPC** - S&P 500 Index of the **New York Stock Exchange (NYSE) and NASDAQ**

The S&P 500 Index tracks the performance of 500 large companies listed on the New York Stock Exchange (NYSE) and NASDAQ. It is widely regarded as one of the best gauges of the U.S. stock market’s health and is a common benchmark for investment performance.

3.  **\^N225** - Nikkei 225 Index of the **Tokyo Stock Exchange**

The Nikkei 225 is a stock market index for the Tokyo Stock Exchange (TSE), Japan’s premier stock exchange. It comprises 225 of the largest and most liquid stocks in Japan, serving as a key indicator of the Japanese economy and stock market trends.

4.  **\^HSI** - Hang Seng Index of the **Hong Kong Stock Exchange (HKEX)**

The Hang Seng Index (HSI) is the main stock market index of the Hong Kong Stock Exchange (HKEX). It tracks the performance of the largest and most liquid companies listed in Hong Kong, providing insight into the broader economic health of Hong Kong and China.

5.  **\^N100** - Euronext 100 Index of the **Euronext Stock Exchange** (covers major European markets)

The Euronext 100 Index represents the largest and most liquid stocks traded on the Euronext Stock Exchange, which operates in several European countries, including France, Belgium, Netherlands, and Portugal. The index includes 100 blue-chip companies, reflecting the performance of major European markets.

Cleaning the Data

# Cleaning Data
BSESN <- na.omit(BSESN$BSESN.Close)
GSPC <- na.omit(GSPC$GSPC.Close)
N225 <- na.omit(N225$N225.Close)
HSI <- na.omit(HSI$HSI.Close)
N100 <- na.omit(N100$N100.Close)

Taking Log Returns of the Chosen Exchanges and Testing Stationarity

Checking Stationarity of the Data:

Used Test: Augmented Dicky Fuller’s Test

H0: The series is not stationary

H1: The series is stationary

  1. BSESN
# Log Differencing and ADF Test
LRBSESN <- diff(log(BSESN))
LRBSESN <- na.omit(LRBSESN)
adf.test(LRBSESN)
Warning in adf.test(LRBSESN) : p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LRBSESN
Dickey-Fuller = -10.311, Lag order = 11, p-value = 0.01
alternative hypothesis: stationary

Results - p-value <0.05 Hence We Reject H0

  1. GSPC
LRGSPC <- diff(log(GSPC))
LRGSPC <- na.omit(LRGSPC)
adf.test(LRGSPC)
Warning in adf.test(LRGSPC) : p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LRGSPC
Dickey-Fuller = -10.808, Lag order = 11, p-value = 0.01
alternative hypothesis: stationary

Results - p-value <0.05 Hence We Reject H0

  1. N225
LRN225 <- diff(log(N225))
LRN225 <- na.omit(LRN225)
adf.test(LRN225)
Warning in adf.test(LRN225) : p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LRN225
Dickey-Fuller = -11.387, Lag order = 11, p-value = 0.01
alternative hypothesis: stationary

Results - p-value <0.05 Hence We Reject H0

  1. HSI
LRHSI <- diff(log(HSI))
LRHSI <- na.omit(LRHSI)
adf.test(LRHSI)
Warning in adf.test(LRHSI) : p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LRHSI
Dickey-Fuller = -11.936, Lag order = 11, p-value = 0.01
alternative hypothesis: stationary

Results - p-value <0.05 Hence We Reject H0

  1. N100
LRN100 <- diff(log(N100))
LRN100 <- na.omit(LRN100)
adf.test(LRN100)
Warning in adf.test(LRN100) : p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LRN100
Dickey-Fuller = -10.877, Lag order = 11, p-value = 0.01
alternative hypothesis: stationary

Results - p-value <0.05 Hence We Reject H0

Normality Test for all Series:

Used Test: Jarque Bera Test

H0: The series is normally ditsributed

H1: The series is not normally distributed

jarque.bera.test(LRBSESN)

    Jarque Bera Test

data:  LRBSESN
X-squared = 31642, df = 2, p-value < 2.2e-16
jarque.bera.test(LRGSPC)

    Jarque Bera Test

data:  LRGSPC
X-squared = 11983, df = 2, p-value < 2.2e-16
jarque.bera.test(LRN225)

    Jarque Bera Test

data:  LRN225
X-squared = 669.81, df = 2, p-value < 2.2e-16
jarque.bera.test(LRHSI)

    Jarque Bera Test

data:  LRHSI
X-squared = 507.86, df = 2, p-value < 2.2e-16
jarque.bera.test(LRN100)

    Jarque Bera Test

data:  LRN100
X-squared = 14220, df = 2, p-value < 2.2e-16

Explanation:

p-value for all 5 series is less than 0.05

hence we reject the null hypothesis and say that all the series are not normally distributed

Implication: all the 5 return series are risky stock exchanges

Plotting all 5 Stock Exchanges

plotNormalHistogram(LRBSESN)

plotNormalDensity(LRBSESN)

plotNormalHistogram(LRGSPC)

plotNormalDensity(LRGSPC)

plotNormalHistogram(LRN225)

plotNormalDensity(LRN225)

plotNormalHistogram(LRHSI)

plotNormalDensity(LRHSI)

plotNormalHistogram(LRN100)

plotNormalDensity(LRN100)

Obtaining Basic Statistics of Log Returns of Each Stock Exchanges

# Print Basic Stats 
basicStats(LRBSESN)
basicStats(LRGSPC)
basicStats(LRN225)
basicStats(LRHSI)
basicStats(LRN100)

ARIMA Modelling of Return Series

1.BSESN

acf(LRBSESN) 

pacf(LRBSESN) 


#Using Auto arima 
arma_pq_LRBSESN = auto.arima(LRBSESN); arma_pq_LRBSESN
Series: LRBSESN 
ARIMA(0,0,1) with non-zero mean 

Coefficients:
          ma1   mean
      -0.0403  5e-04
s.e.   0.0255  3e-04

sigma^2 = 0.000137:  log likelihood = 4468.7
AIC=-8931.39   AICc=-8931.38   BIC=-8915.5
#Application of ARIMA

model2 = arima(LRBSESN,order = c(0,0,1))
model2

Call:
arima(x = LRBSESN, order = c(0, 0, 1))

Coefficients:
          ma1  intercept
      -0.0403      5e-04
s.e.   0.0255      3e-04

sigma^2 estimated as 0.0001368:  log likelihood = 4468.7,  aic = -8931.39
et = residuals(model2)
et
Time Series:
Start = 1 
End = 1475 
Frequency = 1 
   [1] -5.287149e-04 -1.094593e-03  4.643525e-03  5.080465e-03  5.497748e-03  2.334836e-03 -7.145648e-04  1.499500e-03  2.118991e-03  6.803843e-03 -2.322470e-03
  [12]  8.289655e-03  4.893462e-03  6.783782e-03  7.792063e-03  9.306495e-03  4.592551e-04 -3.576005e-03  5.778377e-03 -7.182737e-03 -2.712641e-03 -2.247936e-03
  [23] -2.427464e-02 -1.036010e-02 -1.721056e-02 -4.524536e-03  8.951891e-03 -1.206327e-02  7.628526e-03 -4.429723e-03  3.441676e-03 -8.770656e-03 -7.834094e-03
  [34] -2.936669e-03  3.549798e-03 -1.121307e-03  8.935226e-03  8.698119e-03 -3.053093e-03 -5.375754e-03 -4.749837e-03 -9.561151e-03 -1.371104e-02 -9.630857e-03
  [45]  8.692513e-03 -1.497682e-03  1.759729e-02 -1.610822e-03 -1.201297e-03 -5.012034e-03 -1.595846e-02 -8.808935e-03  1.364786e-03  3.756561e-03 -4.291614e-03
  [56] -1.317895e-02  1.326640e-02  3.279813e-03 -6.602817e-03  7.877313e-03  3.262738e-03 -1.097416e-02  1.638890e-02  1.042860e-03  4.320574e-03  2.369883e-03
  [67]  1.355755e-03  4.263121e-03  2.337158e-03  2.872399e-03  2.210243e-03 -2.270065e-03  2.174879e-03 -7.673369e-04  4.764152e-04  4.307658e-03 -3.679610e-03
  [78]  5.472575e-03  7.056056e-03  5.206807e-03  1.517202e-04 -2.594015e-03 -5.982385e-03  7.594306e-03  2.343656e-05  2.407637e-03 -2.489063e-03  7.565693e-03
  [89]  3.785692e-04 -8.586526e-04 -4.949642e-03 -7.483782e-03 -9.411330e-03 -7.578373e-03  1.938316e-04 -9.386527e-03  8.329437e-03  7.343969e-03  6.646777e-03
 [100] -6.415279e-03 -2.007869e-03  1.125929e-02 -2.757725e-03 -6.758291e-03 -3.895784e-03  7.195447e-03  7.821308e-03 -7.470921e-04  5.774004e-04  5.382762e-03
 [111]  1.007956e-03 -4.380530e-03 -6.433624e-05 -2.593408e-03 -8.003197e-03  6.520776e-03 -3.490698e-03  6.577700e-03 -6.411995e-03 -2.180433e-04 -8.243525e-03
 [122] -5.955787e-03  1.019740e-02 -4.604673e-03  2.532772e-03  7.100199e-03 -2.218305e-03  1.735026e-03  7.289514e-03  8.227966e-03  5.423577e-04  7.266103e-03
 [133] -4.076366e-04 -6.510913e-03  4.609737e-03 -4.349166e-03 -1.300678e-03  3.417695e-03  5.693584e-03  2.610832e-03  4.896663e-04  2.928781e-03  9.081355e-03
 [144]  4.061757e-03  2.636313e-03 -2.670187e-03 -1.016780e-02  9.541463e-03  3.477109e-03 -1.067097e-03  5.312570e-03  3.303758e-03 -4.470029e-03 -6.636198e-03
 [155]  4.704414e-03 -5.316051e-03  6.791805e-03  8.440122e-03  8.055371e-06  8.171293e-04 -2.700428e-03  1.087328e-02  5.143403e-03 -4.783228e-03 -1.555465e-03
 [166] -1.741919e-03 -9.227351e-03 -4.929649e-03 -4.378771e-03  5.196636e-03  3.531376e-03 -1.262886e-02 -1.453757e-02  7.014547e-03  9.599992e-03 -1.347804e-02
 [177] -8.932936e-03 -5.428927e-03 -8.294396e-03 -1.552041e-02  8.373835e-03 -3.177402e-03 -6.628913e-03 -3.456499e-03  7.565702e-03 -1.539646e-02 -2.380693e-02
 [188] -2.425564e-02  1.337472e-03 -5.547381e-03  1.262495e-02 -2.210488e-02  1.990766e-02  4.066516e-03  8.142311e-03 -1.113603e-02 -1.439265e-02 -6.390180e-03
 [199] -9.219929e-03  4.615686e-03 -1.048397e-02 -1.110360e-02  2.034196e-02 -4.883053e-03  1.541357e-02 -1.867254e-04  1.617314e-02 -1.599342e-03  5.929388e-04
 [210]  6.508236e-03 -2.500701e-03 -1.049272e-02  8.539985e-03 -2.419127e-04  2.843357e-03  5.160553e-03  8.613855e-03 -8.599282e-03 -8.635054e-03 -7.097266e-03
 [221]  9.807607e-03  4.369312e-03  5.383819e-03  1.231816e-02  6.416655e-04  8.005440e-04 -3.430747e-03 -7.592715e-03 -1.689692e-02  8.979422e-03 -2.035768e-02
 [232]  4.093886e-03  1.738831e-02  4.385075e-03  5.879737e-04  8.013191e-03  1.928937e-03  3.331964e-03 -1.824815e-03 -1.969826e-02 -8.944984e-03  4.181109e-03
 [243]  4.057438e-03  7.145189e-03 -4.595517e-04 -5.447310e-03 -1.131628e-02  4.124184e-03  3.985955e-03  3.286855e-03  6.044290e-03 -3.214068e-03 -3.324816e-03
 [254] -4.998006e-03  1.216371e-02  5.668560e-05  9.399168e-04 -1.323987e-04  4.752370e-03 -4.002136e-03 -9.942864e-03  1.481237e-03 -5.150738e-03 -1.101320e-02
 [265] -2.760337e-03 -6.609890e-04  1.798281e-02  6.059924e-03  2.831477e-03  5.302169e-04  9.247506e-03 -2.542716e-04 -1.207631e-02 -5.153674e-03 -7.377404e-03
 [276] -8.514276e-03 -2.734400e-03 -9.333873e-03 -5.007188e-03  1.063685e-02  3.879641e-03 -1.107273e-03  8.926774e-03 -6.795412e-03 -2.688263e-03 -1.681535e-03
 [287]  4.877449e-03  1.012864e-02  5.190592e-03  2.129410e-03 -1.900186e-03  9.789852e-03  1.279193e-02  5.752050e-03 -2.110395e-04  6.587741e-03  1.609525e-03
 [298]  6.570904e-03  3.566005e-04 -6.304059e-03 -1.013242e-02  1.024219e-02 -2.735094e-03  1.014331e-02  8.319639e-03  4.562586e-03 -4.938141e-03 -5.674848e-03
 [309]  3.834939e-03 -4.529772e-03  5.451481e-03 -9.424383e-03 -3.330476e-04  3.610165e-03  3.202861e-03  9.074416e-03 -3.601600e-03 -1.338985e-02 -3.134007e-03
 [320]  1.197980e-02 -8.358270e-03  7.798526e-03 -1.116842e-03 -1.844623e-03 -1.055166e-03 -9.915311e-03 -9.335600e-03 -1.370870e-02 -7.177616e-03 -3.360935e-03
 [331] -1.063401e-02  5.177549e-03 -5.778271e-03  6.731024e-03  1.402265e-02  3.685125e-02 -8.807472e-03  2.727172e-03 -8.074678e-03  1.509295e-02  6.376683e-03
 [342]  1.414995e-03 -6.708189e-03  7.532377e-03 -3.172392e-03  1.319645e-02 -4.565194e-03 -1.461155e-02  1.074603e-03  3.775941e-03  3.799694e-03 -5.220777e-03
 [353] -1.113626e-03 -7.865469e-03 -1.336226e-02  1.140575e-03  1.230291e-03  1.195689e-02 -1.036729e-02 -2.758821e-03  7.316923e-03  3.756759e-03 -5.065776e-04
 [364] -5.391203e-03  6.649477e-03  3.022918e-03  1.787366e-04  1.218162e-03 -1.040430e-02 -2.120234e-02 -1.103894e-03 -5.056154e-03  6.158666e-03 -2.506999e-03
 [375]  3.518679e-03  5.633298e-03  1.871838e-03 -8.586005e-03 -1.537367e-02 -9.144627e-03 -2.156186e-03 -4.164489e-03 -1.122911e-03  8.086132e-04 -5.680513e-03
 [386] -8.445132e-03  1.385700e-03 -1.288333e-02  1.661439e-03 -1.178332e-02  6.530445e-03 -8.025922e-03  1.637076e-02  6.940859e-03 -1.697153e-02  8.317771e-03
 [397]  8.595446e-04  9.154505e-04 -2.471040e-03 -7.810013e-03 -1.680713e-02  5.046464e-03  2.106431e-02  4.250505e-03 -5.388729e-03 -1.100841e-02  6.134820e-03
 [408] -2.110530e-02  3.051585e-03 -2.581186e-03  8.545205e-03  4.245625e-03  3.025737e-03 -4.871211e-03  6.826001e-03 -7.264007e-03 -1.825824e-02  1.016789e-03
 [419] -1.342266e-02  5.080366e-02  2.942792e-02  8.522915e-04 -1.344534e-02  9.157935e-03 -4.442593e-03 -4.700395e-03 -1.010787e-02 -6.118307e-03 -1.220386e-02
 [430] -4.764651e-03  1.635813e-02 -7.680235e-03  5.666949e-03  2.002985e-03  7.168162e-03  2.183637e-03  1.124276e-02  6.225750e-03 -8.813212e-03  1.565257e-03
 [441] -1.436243e-03  3.923311e-04  1.911847e-02  5.764050e-03  1.642570e-03  4.476651e-04  2.906705e-03 -1.731729e-03  4.905070e-03  4.218078e-03 -8.498486e-03
 [452] -3.246891e-04 -6.220469e-03  3.474005e-03  1.366479e-03 -2.257767e-03  3.988779e-03  4.131604e-03 -2.231158e-03 -5.936326e-03  1.228836e-02 -1.682510e-03
 [463]  4.288123e-03  2.325344e-03 -8.632696e-03 -6.573713e-04 -3.651792e-03  3.627431e-03 -2.100807e-03 -8.834318e-03  1.743676e-04 -6.640712e-03  3.500225e-03
 [474]  3.802873e-03  1.012984e-02 -1.839354e-03  9.459882e-03  4.845092e-03  2.452215e-03 -2.331816e-04 -1.457290e-03 -4.938979e-03 -7.914848e-03  9.110654e-03
 [485] -5.601586e-04 -7.885570e-03  8.166353e-03 -4.085907e-03 -1.986583e-02  3.414848e-03 -1.643728e-03  1.484696e-02  3.632017e-03  5.861507e-03  1.939109e-03
 [496] -2.342917e-03  8.187310e-04 -1.762268e-04 -1.050007e-02 -5.888570e-03 -5.808430e-03  5.821429e-03  5.184673e-03 -1.137475e-02 -5.557730e-03  4.903776e-03
 [507] -7.255039e-03 -5.469759e-03 -2.185795e-02  2.134468e-02  8.968720e-03  3.809472e-03 -4.343920e-03 -4.640637e-03  5.053453e-03  8.138953e-03 -2.742989e-03
 [518] -5.510499e-03 -5.645843e-03 -4.678929e-03  9.723492e-03 -3.829581e-03 -2.046247e-02 -3.373174e-03 -1.043577e-02 -4.533939e-03 -3.781884e-02 -6.047894e-03
 [529]  1.173881e-02 -5.603692e-03  8.498627e-04 -2.399296e-02 -5.453616e-02 -9.600785e-04 -8.586933e-02  3.566471e-02 -8.198623e-02 -2.999168e-02 -5.925297e-02
 [540] -2.324115e-02  5.449563e-02 -1.393374e-01  2.018948e-02  6.776669e-02  5.047718e-02 -2.871906e-03 -4.785390e-02  3.307179e-02 -4.086910e-02 -2.630809e-02
 [551]  8.437325e-02 -2.895528e-03  4.083520e-02 -1.405566e-02 -1.124000e-02  6.339651e-03  3.145545e-02  2.626928e-03 -3.288487e-02  2.211838e-02  1.566761e-02
 [562] -1.684431e-02  1.199429e-02  1.160184e-02  1.863569e-02  3.026498e-02 -6.051566e-02 -1.124203e-02  6.389029e-03 -7.935957e-03  5.484650e-03 -2.872097e-03
 [573] -6.671846e-03  1.933382e-02 -2.779741e-02 -2.442917e-03 -3.558500e-02  3.604359e-03  2.003407e-02  3.993768e-03 -8.804801e-03 -2.934802e-03  3.138551e-02
 [584]  1.941173e-02  7.184249e-03  2.653575e-02  1.610676e-02  8.495162e-03 -3.956970e-03  8.306470e-03  2.247509e-03 -1.253917e-02  7.494834e-03 -2.112313e-02
 [595]  5.839701e-03 -1.675781e-02  1.007480e-02 -3.008454e-03  2.004327e-02  1.548516e-02  5.266374e-03  1.445729e-02 -1.590589e-02 -1.926426e-03  8.810787e-03
 [606] -6.141397e-03 -2.070576e-03  1.358235e-02  1.208018e-02  4.917814e-03  1.253328e-02  5.108527e-03 -9.774549e-03  1.027825e-02 -4.010754e-03  2.035359e-03
 [617] -1.860077e-02 -7.435837e-04  1.103438e-02  1.485573e-02  1.079975e-02  1.349295e-02 -1.522947e-03  6.500489e-03 -5.563176e-04 -5.642527e-03  1.386608e-02
 [628] -1.097514e-02 -9.796514e-03 -4.338348e-03 -1.859253e-02  1.879170e-02 -4.103774e-04  9.037530e-03  2.468403e-04  3.208255e-03  5.488295e-03 -1.267482e-03
 [639] -2.108276e-03 -1.197027e-02  3.571800e-03  1.210109e-02  2.214416e-03 -1.069170e-02  4.646741e-03  9.107735e-03  1.006110e-03  5.430593e-03  7.155937e-04
 [650]  8.519938e-03 -2.165940e-02  5.642968e-03  4.462864e-03 -2.770770e-03 -1.701391e-02  3.644469e-04 -1.851345e-03 -5.067621e-03  1.606371e-02  4.985416e-04
 [661] -3.018023e-03  6.760164e-03  6.356402e-03 -8.510903e-03 -4.301859e-03 -2.180424e-02 -9.313395e-03 -2.631265e-03 -3.066308e-02  2.083815e-02  1.605964e-02
 [672] -8.939057e-05  1.972743e-03  1.595595e-02  7.251535e-03  1.507704e-02  7.754337e-03  7.384843e-03  7.883075e-03  1.881909e-03  3.419586e-04  3.655942e-03
 [683] -2.685402e-02  4.791138e-03  1.083607e-02  2.706955e-03  3.605005e-03 -4.032081e-03  2.449591e-03 -1.377746e-02  8.267632e-03 -1.509019e-02 -5.455413e-03
 [694] -4.156147e-03  2.934020e-03  1.218945e-02  8.755278e-03  1.750678e-02  1.347606e-02  1.670178e-02  1.600031e-02  7.405371e-03 -5.655887e-03  1.234673e-03
 [705]  1.119959e-02  5.095377e-03 -1.352659e-02  5.394202e-03  4.134098e-03  9.716664e-03 -1.585454e-02  8.647016e-03 -2.655330e-03  1.076797e-02 -9.189758e-04
 [716] -2.243338e-04  9.439368e-03  7.542675e-03  3.777466e-03  1.043194e-02 -3.214632e-03  2.378530e-03  2.925867e-03 -1.869419e-04  8.157273e-03  4.599767e-03
 [727]  1.169760e-03 -3.088091e-02  8.130916e-03  9.277054e-03  1.119220e-02  7.997606e-03  5.268842e-03  2.489889e-03 -3.074230e-04  1.933741e-03  5.973062e-03
 [738]  5.128368e-03 -5.767611e-03 -2.424355e-03  1.361625e-02  9.963358e-03  4.903220e-03 -8.179787e-04  1.306274e-03 -1.160589e-02 -1.062167e-02  1.608526e-02
 [749]  8.074064e-03 -3.556362e-03 -1.580944e-02 -1.207344e-02 -2.058561e-02 -1.270462e-02 -1.366261e-02  4.773658e-02  2.574088e-02  9.677743e-03  6.984006e-03
 [760]  2.082193e-03  1.166057e-02 -4.286584e-04 -9.155940e-04  3.768286e-03 -1.149470e-04  1.124237e-02 -1.020275e-03 -8.268988e-03 -8.207590e-03 -9.355436e-03
 [771] -2.365694e-02 -1.324913e-03  1.992902e-02  5.348099e-03 -3.903662e-02  1.306968e-02  8.939622e-03  2.240850e-02 -1.131570e-02 -9.676677e-03 -1.954085e-04
 [782]  1.099682e-02  4.894334e-03 -9.868486e-03 -8.759018e-03 -1.485107e-03 -1.180283e-02 -1.280824e-02  1.192389e-02 -1.779948e-03  5.026626e-03 -1.787026e-02
 [793] -1.639928e-02  1.049019e-02  2.266486e-02 -1.219530e-02  9.456081e-03 -1.768681e-02 -3.715674e-04  8.783701e-03  1.538083e-03 -3.571234e-03 -3.570590e-02
 [804]  1.175069e-02  5.292431e-03  2.791995e-04 -1.874319e-02 -6.363284e-03  7.056295e-03 -4.445300e-03  9.861874e-03  1.134103e-02  1.594795e-02  7.728218e-04
 [815] -2.044566e-02 -2.647626e-03 -1.021196e-02  7.823383e-03  5.376912e-03  1.092870e-02 -6.978860e-03 -1.042272e-02 -7.736849e-05  1.673724e-02  1.243920e-02
 [826] -5.821936e-03 -7.540985e-03  1.867411e-02  2.439455e-03 -7.001453e-04  6.933169e-03  1.677696e-03  5.553775e-03  9.665718e-03 -1.747785e-04 -2.167432e-03
 [837]  6.756655e-03 -2.780232e-03  3.748786e-03 -1.375954e-03 -6.978478e-03  6.088797e-03  3.057500e-03  1.070342e-03  3.734826e-03 -5.514058e-03 -4.145344e-03
 [848] -2.780630e-04  3.858668e-03 -8.829667e-05 -5.907154e-03  6.731300e-03  4.036459e-03 -3.938142e-03 -4.205228e-03 -1.958889e-03 -3.725414e-03  2.504450e-03
 [859]  7.090191e-03 -5.851317e-04  3.117127e-03 -9.588310e-03 -4.383293e-03 -9.489640e-04  6.999521e-03  2.309357e-03  4.383021e-03 -6.917743e-04 -1.164384e-02
 [870] -7.759413e-03  1.133452e-02  2.561293e-03 -2.746119e-03 -5.813682e-03 -3.320682e-03  3.335695e-03 -1.639023e-03  6.313344e-03  1.608716e-02  1.023390e-02
 [881]  2.158465e-03 -4.383283e-03  1.611535e-03  2.336765e-03 -9.473702e-04  5.263217e-03  1.045728e-02  2.523793e-03  3.352449e-03 -3.301586e-03 -6.058119e-03
 [892]  3.326073e-03  6.850453e-03 -5.027869e-04 -4.475353e-04  2.601232e-03  1.312911e-02  1.159432e-02 -3.776184e-03  8.263305e-03  4.601747e-03  2.538706e-03
 [903] -7.114884e-04 -1.044838e-03  3.836925e-04 -2.685129e-03  5.681110e-04  7.648911e-03  6.885605e-03 -2.357793e-03 -9.544703e-03  7.855967e-03 -1.520118e-03
 [914]  1.555118e-02  2.831575e-03  8.898593e-05 -7.363636e-03 -5.082847e-03 -5.560158e-03 -6.859185e-03  8.250627e-03  7.303206e-03 -9.556007e-03  7.312960e-03
 [925]  6.147634e-03  1.009517e-03  1.992774e-03  7.047608e-03  9.092108e-03  7.320982e-03 -1.022239e-03 -7.973447e-03 -6.343299e-03 -2.443807e-03  1.775110e-03
 [936]  5.822656e-03 -3.658897e-03 -1.979329e-02 -1.267519e-02  1.289826e-02 -1.816087e-03 -4.880579e-03  4.223673e-03  7.581405e-03 -2.063588e-03 -1.932876e-03
 [947] -7.795035e-03  1.189060e-02  4.916440e-04 -7.043775e-03 -6.018051e-03 -6.980805e-03 -2.061193e-02  2.043527e-03 -5.959344e-03  6.998740e-03 -2.936189e-02
 [958]  9.859134e-04 -3.898683e-03  1.013309e-02  1.326468e-02 -1.314947e-02 -1.763483e-02  1.427640e-02  1.753579e-02  2.872483e-03 -7.469474e-04 -9.142251e-03
 [969] -3.740813e-03 -6.343438e-03  1.185232e-03 -1.594687e-02 -2.224584e-02  7.453297e-03  1.058555e-02  6.646627e-03 -3.584557e-03  4.508017e-03  7.943891e-03
 [980] -1.767627e-03 -7.964066e-04  7.372393e-03  1.561056e-02  1.141644e-02  6.061368e-03 -1.064096e-02  1.449936e-03  1.038080e-02  3.560203e-03  8.385599e-03
 [991]  1.216340e-03 -6.661046e-04  8.602109e-04 -9.558177e-03 -1.175649e-02 -1.159685e-02 -8.195860e-03 -2.737487e-02  4.739824e-03 -1.041999e-02
 [ reached getOption("max.print") -- omitted 475 entries ]
#Checking of significance of ARMA
coeftest(model2)

z test of coefficients:

             Estimate  Std. Error z value Pr(>|z|)  
ma1       -0.04027260  0.02549300 -1.5798  0.11416  
intercept  0.00051470  0.00029342  1.7541  0.07941 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))

Explanation:

  1. Auto ARIMA: gave order of ARIMA (0,0,1)

    i.e. no autoregressive (AR) terms, no differencing (D) terms, and one moving average (MA) term.

  2. Log likelihood: 4468.7 This is a measure of the model’s goodness of fit. A higher log likelihood indicates a better fit.

    AIC: -8931.39 The Akaike Information Criterion (AIC) is a measure of model complexity and fit. A lower AIC indicates a better model.

    sigma^2: 0.000137 This is the estimated variance of the residuals, representing the model’s ability to fit the data. A smaller value suggests a better fit.

    Implication: The ARIMA(0,0,1) model provides a parsimonious representation of the LRNSEI time series. The model suggests that the current value of the series is influenced by the previous period’s error.

  3. The significance test indicate that the MA1 coefficient in the ARIMA(0,0,1) model is not statistically significant. The intercept coefficient is close to being statistically significant, but more evidence is needed to confirm its significance.

2.GSPC

acf(LRGSPC) 

pacf(LRGSPC) 


#Using Auto arima 
arma_pq_LRGSPC = auto.arima(LRGSPC); arma_pq_LRGSPC
Series: LRGSPC 
ARIMA(4,0,4) with zero mean 

Coefficients:
          ar1     ar2      ar3      ar4     ma1      ma2     ma3     ma4
      -0.2431  0.7515  -0.3713  -0.8419  0.1408  -0.7099  0.4569  0.6795
s.e.   0.0391  0.0390   0.0352   0.0371  0.0510   0.0483  0.0442  0.0483

sigma^2 = 0.0001553:  log likelihood = 4476.69
AIC=-8935.38   AICc=-8935.26   BIC=-8887.52
#Application of ARIMA

model2 = arima(LRGSPC,order = c(4,0,4))
model2

Call:
arima(x = LRGSPC, order = c(4, 0, 4))

Coefficients:
          ar1     ar2      ar3      ar4     ma1      ma2     ma3     ma4  intercept
      -0.2399  0.7515  -0.3747  -0.8428  0.1366  -0.7098  0.4611  0.6797      4e-04
s.e.   0.0400  0.0385   0.0350   0.0368  0.0521   0.0478  0.0442  0.0483      3e-04

sigma^2 estimated as 0.0001543:  log likelihood = 4477.52,  aic = -8935.05
et = residuals(model2)
et
Time Series:
Start = 1 
End = 1508 
Frequency = 1 
   [1]  5.707720e-03  4.474868e-03  6.440520e-03  1.912147e-03  1.054237e-03 -1.431505e-03  7.258731e-03  6.769181e-03 -2.868039e-03  7.186766e-03  3.139342e-05
  [12]  2.530853e-03  8.455008e-03  3.202811e-03 -2.061236e-03  1.854117e-03  1.033949e-02 -4.733918e-03 -1.374823e-02 -2.494570e-04 -1.167802e-04 -2.300643e-02
  [23] -4.359253e-02  1.278865e-02 -7.227884e-04 -4.197355e-02  8.985966e-03  1.876801e-02 -8.408609e-04  1.402986e-02  1.298726e-02  1.039657e-03 -5.038338e-03
  [34] -6.191760e-03  1.636426e-03  1.473866e-02  1.253113e-02 -1.465180e-02 -1.385204e-02 -1.361624e-02  4.908355e-03  1.265360e-02  3.823551e-03 -2.696318e-03
  [45]  4.009981e-03  1.542701e-02  2.870237e-04 -8.117837e-03 -5.260534e-03  3.814538e-04  1.478532e-03 -1.347109e-02 -2.708782e-03 -1.534526e-03 -2.811361e-02
  [56] -2.489394e-02  2.677148e-02 -1.415586e-02 -5.775201e-03  1.293320e-02 -1.924192e-02  4.511956e-03  1.719163e-02  3.526988e-03 -2.142545e-02  1.549976e-03
  [67]  1.692421e-02 -9.814699e-04  2.345743e-03  1.753867e-03  2.845362e-03  1.292431e-02  3.568157e-04 -6.941345e-03 -7.125672e-03 -1.979683e-03 -1.093711e-02
  [78] -2.039250e-03  1.136876e-02 -3.091581e-05 -1.061170e-02  2.105251e-03 -7.452651e-03 -1.261776e-03  1.213169e-02  6.447311e-03 -3.206747e-03  9.484166e-03
  [89]  8.427695e-03  1.548156e-03  6.068496e-04 -6.277624e-03  4.446262e-03  7.234967e-06 -2.718776e-03  5.055548e-03 -2.587446e-03 -2.004629e-04 -1.283866e-03
 [100] -3.108915e-03 -1.138378e-02  1.300844e-02 -5.919715e-03  1.007012e-02  2.691984e-03  1.737369e-03  3.979368e-03  3.535451e-03 -3.613868e-04  5.044379e-03
 [111]  6.527152e-04 -3.237778e-03  1.406429e-03 -2.702257e-03 -2.495704e-03 -6.918300e-03  2.864279e-03 -7.536536e-03  3.051006e-03 -1.437193e-02  1.560996e-03
 [122] -9.735825e-03  5.170526e-03 -1.300550e-03  3.868809e-03 -7.628035e-03  1.079933e-02  7.587837e-03  1.195860e-02  1.581950e-03 -5.698056e-03  5.394895e-03
 [133]  2.939878e-03 -2.722868e-03  3.375912e-03  3.570297e-03 -5.374985e-03  1.983132e-05  3.359034e-04  5.636321e-03  7.602988e-03 -2.406134e-03 -8.529023e-03
 [144] -6.174529e-03  5.023158e-03 -7.036354e-04  4.957045e-03  3.620414e-03  3.960162e-03  7.303845e-04  6.691002e-04 -2.897645e-03 -6.053894e-03 -5.040274e-03
 [155]  6.640502e-03 -7.342768e-03  5.265191e-03  3.740043e-03  1.258457e-03  1.084936e-03  9.666217e-04 -2.761975e-03  7.957530e-03  7.116152e-03  1.140508e-03
 [166]  3.359470e-03 -4.188084e-03 -1.709977e-03 -1.634210e-03 -2.162023e-03 -4.650344e-03 -1.116016e-03 -1.505225e-05  4.298691e-03 -1.649952e-03  4.793003e-03
 [177] -1.708601e-04 -5.575453e-03  4.993061e-03  2.817415e-03  7.431116e-03 -3.053739e-04 -4.514974e-03 -3.474568e-03 -2.799469e-03  8.844048e-04  1.595613e-03
 [188]  2.645362e-03  3.630077e-04  5.801248e-05 -9.073879e-03 -6.457541e-03 -2.102129e-03 -6.157565e-04 -3.535386e-02 -2.344826e-02  1.254498e-02 -2.913489e-03
 [199]  1.680044e-02  2.217432e-03 -1.707283e-02 -3.172243e-03 -4.420339e-04 -8.401898e-03 -2.853060e-02  1.281899e-02 -1.366475e-02 -1.185638e-02  1.235668e-02
 [210]  1.531224e-02  5.007243e-03  2.568507e-04  7.356833e-04  1.209616e-02  1.947066e-02 -6.419972e-04 -1.138080e-02 -2.384717e-02 -9.676005e-04 -1.033159e-02
 [221]  1.340067e-02 -1.685921e-05 -1.397061e-02 -2.436723e-02  4.638434e-03 -8.617446e-03  1.510417e-02  3.200562e-03  2.247316e-02 -1.495892e-03  8.526609e-03
 [232]  1.032677e-02 -2.866666e-02 -8.696363e-03 -1.866208e-02 -3.979372e-03 -3.485495e-04  4.968626e-03 -4.908587e-03 -1.560868e-02 -2.778784e-02  4.972849e-03
 [243] -1.762085e-02 -1.507365e-02 -2.511117e-02 -2.943805e-02  4.289212e-02  1.402942e-02 -4.870162e-03  6.939945e-03  7.546433e-03 -2.843553e-02  3.762468e-02
 [254]  7.363153e-03  1.067254e-02 -2.213687e-03  7.574139e-03 -5.808214e-03  2.643329e-04  7.858096e-03  8.486693e-03  5.586314e-03  1.361400e-02 -1.477029e-02
 [265] -3.662303e-03  3.198659e-03  6.702565e-03 -5.632257e-03 -1.429224e-03  1.561727e-02  1.178135e-02 -2.505235e-03  6.923647e-03  2.440547e-03 -1.536817e-03
 [276] -1.036617e-02  1.397970e-03  1.919351e-03  1.298390e-02  3.239340e-03 -4.422664e-03  8.281555e-03  2.782948e-03  2.482800e-04 -1.927262e-03  6.774510e-03
 [287]  2.405733e-03 -5.662687e-04 -2.911328e-03 -2.878412e-03  3.654289e-03 -2.439766e-03 -3.155423e-03 -5.375296e-03 -8.049959e-03 -3.117342e-03  1.546585e-02
 [298]  1.871172e-03  6.007134e-03 -2.911397e-03  5.580023e-03  3.113294e-03  2.684968e-03 -4.020203e-03  1.234682e-02 -1.981848e-02 -3.902838e-03  5.774852e-03
 [309] -3.843752e-03 -1.169435e-04  9.680557e-03  1.050763e-02  2.614646e-03  1.117567e-03  2.187449e-03  4.717952e-03 -6.148132e-04 -5.698435e-03  7.518769e-04
 [320]  2.022827e-03  5.251256e-03  6.984025e-04 -1.047823e-04 -3.107938e-03  1.782331e-03 -7.561894e-04  9.667064e-03 -3.377315e-03  1.798176e-05  3.401151e-03
 [331]  2.958935e-03 -6.357913e-04 -6.700963e-03 -4.160235e-03  9.452211e-03 -3.965685e-03 -1.899250e-02 -2.823648e-03 -2.828647e-03  3.373559e-03 -2.479297e-02
 [342]  5.209154e-03  6.350087e-03  9.557394e-03 -8.686851e-03 -5.060920e-03  4.620294e-03  2.204269e-03 -1.591732e-02  2.472806e-03 -9.444295e-03 -8.599842e-03
 [353]  6.678511e-04 -1.374290e-02 -5.123291e-03  2.130499e-02  1.078118e-02  4.612509e-03  1.145506e-02  5.090546e-03 -6.000353e-04 -1.912140e-03  3.466069e-03
 [364] -9.675237e-04 -3.466700e-04  9.340189e-03  3.369640e-03  8.211505e-03 -2.715848e-04 -2.531324e-03 -9.531000e-03 -1.340795e-03  3.106992e-03  6.646044e-03
 [375]  5.402125e-03  3.649972e-03  5.486637e-03 -2.770997e-04 -5.551262e-03  1.668850e-03  5.792187e-03  1.594385e-03  4.409714e-03 -1.622584e-03 -4.144831e-03
 [386] -8.195656e-03  3.803271e-03 -5.507437e-03  2.729739e-03  6.869276e-03  5.195775e-03 -7.435326e-03  6.557282e-03 -2.262651e-03 -2.317479e-03 -1.193782e-02
 [397] -8.197514e-03 -8.972317e-03 -2.962284e-02  7.492745e-03  3.246691e-03  1.555770e-02 -7.166144e-03 -1.197757e-02  1.117278e-02 -2.157612e-02 -5.565144e-03
 [408]  2.023111e-02  1.013535e-02 -1.038840e-02  7.131964e-03 -3.399321e-03 -2.263234e-02  5.187734e-03  5.203919e-03  2.345468e-03  1.424612e-02 -8.021781e-05
 [419] -1.163873e-02  1.159097e-02  1.144862e-02  5.298815e-03 -2.054118e-03  3.790242e-03  5.743770e-03  3.702962e-03 -2.827978e-03 -5.355142e-03  1.989042e-03
 [430] -4.545553e-04  1.601444e-03 -5.300323e-03  1.089233e-03 -9.472796e-03  5.144056e-03 -3.747027e-03 -6.626960e-03  2.432688e-03 -1.030385e-02 -2.075912e-02
 [441]  8.532659e-03  1.577403e-02 -4.395625e-03 -1.828494e-02  6.318085e-03  7.391206e-03  1.019684e-02 -6.762844e-04  1.041888e-02 -1.385630e-03  3.807558e-03
 [452] -5.707938e-03  7.829366e-03 -6.210084e-03  3.501341e-03 -7.138928e-04  6.303116e-03  3.749585e-03  1.833245e-03  1.354661e-03 -1.649779e-03  7.979109e-03
 [463]  4.021649e-03 -1.609303e-03 -1.433206e-03  4.409873e-03  1.102620e-03 -4.304367e-05 -1.279471e-04  1.658303e-03 -4.364741e-04  6.946444e-03  7.200612e-04
 [474] -2.054536e-03 -3.461424e-03 -2.072818e-03  2.465197e-03  7.824482e-03  1.938439e-03  3.471090e-03 -5.057282e-03 -9.337057e-03 -8.183944e-03  7.045956e-03
 [485]  1.711027e-03  9.162793e-03 -3.727517e-03 -2.023690e-03  1.293697e-03  9.773371e-03 -7.330344e-04  8.061970e-03 -1.222648e-05  2.968255e-04  2.914332e-03
 [496]  6.150043e-03 -7.149831e-04 -1.151213e-04  4.065946e-03  7.604074e-04 -6.296622e-03  2.239405e-03  9.198829e-03 -7.359097e-03  1.914799e-03 -2.921373e-03
 [507]  4.201020e-03  6.097554e-03 -1.468841e-03  4.684633e-03  9.170153e-04 -1.091738e-04  8.977430e-03  4.188305e-03 -3.982713e-03  2.625645e-04  1.336326e-04
 [518] -8.256945e-03 -1.785372e-02  9.392576e-03  1.242448e-04  1.661727e-03 -1.905092e-02  5.395166e-03  1.471810e-02  1.409929e-02  8.260690e-04 -2.562268e-03
 [529]  4.018504e-03  5.206937e-03  4.249961e-03 -8.195401e-04  7.486456e-04 -4.244627e-03  5.385649e-03 -5.361941e-03 -9.199935e-03 -3.724589e-02 -3.172846e-02
 [540] -7.588930e-03 -4.335368e-02 -1.667213e-02  4.556581e-02 -2.516200e-02  3.180352e-02 -2.579813e-02 -2.320645e-02 -7.908568e-02  4.805645e-02 -4.979797e-02
 [551] -9.978615e-02  6.320211e-02 -9.919201e-02  1.882121e-02 -2.921307e-02 -5.331425e-03 -5.086975e-02 -1.156892e-02  5.991370e-02  5.271767e-02  2.200256e-02
 [562] -5.800133e-03  9.862792e-03 -2.215034e-03 -4.122872e-02  1.015901e-02  8.845326e-03  4.270324e-02  1.971991e-02  1.132959e-02  2.234407e-02 -8.963196e-03
 [573]  2.588165e-02 -2.349726e-03 -6.862274e-03  4.008493e-02 -2.652310e-02 -3.575712e-02  1.514881e-02  7.241313e-04  1.430535e-02  1.580704e-02  3.488436e-03
 [584]  2.173817e-02 -3.561488e-05 -3.761522e-02  1.846483e-03  7.693470e-03 -9.508733e-03  1.143567e-02  1.792936e-02  4.693338e-03 -2.223507e-02 -1.708972e-02
 [595]  8.506300e-03  5.385224e-03  2.694210e-02 -9.400969e-03  1.330611e-02 -6.212698e-03  5.931480e-03  1.034195e-02  2.285157e-02 -8.386997e-03  7.716987e-03
 [606] -2.924704e-03  1.065141e-02  1.069606e-02  7.435035e-04  2.502317e-02  1.698067e-02 -7.052795e-03 -9.016674e-03 -5.911116e-02  2.300335e-03  1.448713e-02
 [617]  1.616279e-02 -5.103636e-03  2.935456e-03 -1.076232e-02  1.342734e-02 -7.072387e-04 -1.969775e-02  1.708496e-03 -1.881636e-02  7.015852e-03  1.687368e-02
 [628]  7.267570e-03 -1.065710e-03  2.320996e-02 -1.512598e-02  1.271773e-02 -8.984902e-03  1.320945e-02 -1.330674e-02  1.373773e-02  6.490001e-03 -6.201008e-04
 [639] -9.441874e-04  1.294627e-02  6.840884e-04  7.344461e-03 -1.323436e-02 -8.541640e-03  6.034749e-03 -6.294683e-03  9.919822e-03 -1.826749e-03  6.683854e-03
 [650]  7.821235e-03  5.715856e-03  3.883634e-03  9.092326e-03 -2.343439e-03  3.972591e-03 -9.696943e-03  1.396229e-02 -8.129750e-04 -6.471242e-04  1.693988e-03
 [661]  3.577349e-03 -6.763531e-03  3.715741e-03  1.900996e-03  1.115652e-02  2.936206e-03  1.091513e-02  1.563111e-03  7.211360e-03 -2.562651e-03  7.790162e-03
 [672]  1.471008e-02 -3.374232e-02 -1.468716e-02 -2.586970e-02  1.680708e-02 -1.659007e-02 -6.740870e-04  7.382039e-03  1.210061e-02 -1.311330e-02 -1.483248e-03
 [683] -1.751637e-02 -6.815865e-03  6.508060e-03 -2.136788e-02 -2.530993e-03  1.488816e-02  1.832252e-02 -8.899697e-03  1.194169e-02  3.064954e-03 -3.807099e-03
 [694]  1.314950e-02 -8.331355e-03  1.083073e-02  9.745211e-03  8.066505e-03  1.283475e-02  9.384404e-07 -1.226884e-02  4.745212e-03 -2.609717e-03 -1.436531e-02
 [705]  1.118885e-04 -2.807556e-03  3.297550e-03  8.606841e-04 -1.589944e-02 -7.489278e-03 -3.041823e-02  6.346231e-03 -8.145923e-03  8.467570e-03  1.409484e-02
 [716]  2.516241e-02  1.310961e-02  8.051662e-03  6.501437e-03  8.243892e-03  4.904457e-03 -6.961192e-03  1.023287e-02  9.190652e-03 -4.274615e-03 -1.839216e-02
 [727]  8.133562e-03 -8.254238e-03  1.026470e-02  1.502545e-02  1.672646e-03 -2.998565e-03 -5.046019e-03  8.156186e-03  2.409254e-03  8.583614e-04  8.241936e-03
 [738]  2.945919e-03 -2.449202e-04 -5.987261e-03 -5.562012e-03 -2.187688e-03 -5.714339e-03  1.075467e-02  5.199342e-03  4.219208e-03 -9.434361e-04 -4.813769e-03
 [749] -2.860043e-03  6.088289e-04  9.443094e-04  9.148223e-03 -3.825891e-03  1.756501e-03  6.188143e-03 -1.254281e-02  4.546922e-03  7.879203e-03  1.372679e-02
 [760]  4.357450e-03 -6.472814e-03 -3.305826e-03  5.663605e-03 -5.114846e-03 -5.210862e-03  6.475873e-03  1.499670e-02 -5.390960e-04 -5.750199e-03  3.107844e-03
 [771] -1.668978e-03 -2.513524e-02  7.486729e-03 -1.598701e-02  1.252468e-02  1.392702e-02  1.861250e-03  4.280086e-03  9.474454e-03  2.981212e-03  5.064306e-03
 [782] -2.092092e-03  3.880832e-03  4.135441e-03 -2.687487e-03 -1.363423e-03 -7.820927e-03 -1.042494e-03 -9.077366e-03  3.251713e-03  1.126525e-02 -2.246999e-02
 [793] -1.016585e-02  2.367434e-02 -7.056391e-03 -1.834927e-02 -1.291824e-02  1.798565e-02 -1.367104e-03  1.285250e-02  6.394348e-03  1.149394e-02 -2.953068e-03
 [804]  9.277186e-03 -5.489908e-03  6.974323e-03 -1.748123e-02  1.456250e-03  5.051265e-03 -5.654320e-03 -1.041088e-02  5.231081e-03  1.532547e-02  2.738370e-04
 [815] -3.563279e-03  3.512864e-03  1.438152e-02  1.386591e-02  7.960886e-04 -2.005857e-03  4.604852e-03  6.822579e-03  2.948011e-04  3.685267e-03 -3.939740e-03
 [826]  1.172574e-02  3.455549e-03 -4.988741e-03 -1.036233e-02  9.941172e-03 -1.026289e-02  1.063330e-02  2.142031e-03  9.722556e-04 -3.141234e-03  8.788587e-03
 [837] -9.523588e-03  3.134675e-03 -8.550123e-03  1.177046e-03  6.436586e-03  9.697604e-03 -1.235048e-02 -8.947491e-03 -2.331572e-02  1.099986e-02  1.572605e-02
 [848] -1.688364e-03 -1.249939e-02 -2.211031e-03  8.627452e-03  2.131737e-03  8.178695e-03 -4.423552e-04  8.853049e-04 -4.823956e-05  1.791815e-03 -3.301276e-03
 [859]  3.268948e-03 -5.873255e-03  1.008559e-02 -1.279465e-03  4.095413e-04 -3.215919e-03  5.008795e-03  8.985508e-04  2.252081e-03 -3.275111e-03 -5.023810e-03
 [870] -1.828067e-03 -1.245882e-02  1.164659e-02  6.839271e-03 -1.945541e-03  3.347842e-03  5.613963e-03 -1.262062e-04  2.731062e-03 -1.825647e-04  6.551304e-03
 [881]  6.937656e-03 -2.124600e-03  1.999820e-03 -9.158009e-03  1.046411e-02  4.098650e-03 -2.429607e-03 -1.313224e-03 -7.876479e-04 -1.077432e-02 -1.536106e-02
 [892]  1.171945e-02  1.060772e-02  9.388631e-04  8.365350e-03  5.070537e-03 -7.105167e-03  1.806045e-03  3.145314e-03 -3.997797e-03 -4.215254e-03  7.635688e-03
 [903] -4.761821e-03  3.565075e-03  2.942739e-03 -1.399879e-03  1.071224e-03  2.331468e-03  3.202349e-03  2.253955e-03  7.281269e-04 -7.241773e-03 -1.273570e-02
 [914]  4.124732e-04  9.187997e-03  8.659820e-03  1.965512e-03  1.015346e-03 -6.042170e-03  7.526304e-03  5.119402e-03 -1.131005e-03 -1.094909e-03  4.157438e-03
 [925] -1.717134e-03 -2.817675e-03 -3.205191e-03 -4.347725e-03 -9.281429e-03  1.455734e-03 -5.478452e-03  6.927013e-03 -7.168525e-04 -1.019900e-02 -1.907428e-02
 [936] -1.166364e-03  8.577817e-03  1.390275e-02 -4.492299e-04 -2.502100e-03 -2.269342e-02  8.728076e-04 -1.136533e-02  1.101621e-02 -1.326799e-02  9.217382e-03
 [947]  1.940523e-03  1.037642e-02 -6.014579e-03 -2.815955e-03 -6.715206e-03  7.858693e-03  1.417115e-02  1.052286e-02  4.513025e-04  6.362919e-03  4.261844e-03
 [958]  1.792070e-03  1.394294e-03  3.960132e-03  4.048548e-03 -6.313119e-03  8.391877e-03  1.945808e-03 -2.412179e-04  3.233661e-03  7.087738e-03  4.747562e-03
 [969]  5.193879e-03  7.077379e-04 -3.158284e-03 -1.018377e-02 -6.732840e-04  6.167597e-03  1.340426e-04  2.643801e-03 -1.803836e-03  3.143880e-03 -8.560109e-04
 [980] -3.060216e-03 -4.413531e-05  2.868418e-03 -2.513242e-02  1.072741e-02 -1.770818e-02 -1.398441e-02  1.200681e-02 -3.627510e-03  5.310940e-03  2.496222e-02
 [991]  1.162753e-03 -7.713912e-03  9.086411e-03 -7.917080e-03 -7.290221e-03  1.459048e-02 -5.021091e-03 -1.603404e-02 -1.136537e-02  1.494777e-02
 [ reached getOption("max.print") -- omitted 508 entries ]
#Checking of significance of ARMA
coeftest(model2)

z test of coefficients:

             Estimate  Std. Error  z value  Pr(>|z|)    
ar1       -0.23989886  0.03995719  -6.0039 1.926e-09 ***
ar2        0.75146942  0.03853706  19.4999 < 2.2e-16 ***
ar3       -0.37472576  0.03497353 -10.7146 < 2.2e-16 ***
ar4       -0.84276029  0.03682745 -22.8840 < 2.2e-16 ***
ma1        0.13658466  0.05206068   2.6236  0.008701 ** 
ma2       -0.70983378  0.04782940 -14.8410 < 2.2e-16 ***
ma3        0.46110583  0.04417324  10.4386 < 2.2e-16 ***
ma4        0.67974688  0.04829892  14.0737 < 2.2e-16 ***
intercept  0.00037583  0.00029508   1.2736  0.202794    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))

Explanation:

  1. Auto ARIMA: gave order of ARIMA (4,0,4)

    i.e. no autoregressive (AR) terms, no differencing (D) terms, and one moving average (MA) term.

  2. Log likelihood: 4476.69 This is a measure of the model’s goodness of fit. A higher log likelihood indicates a better fit.

    AIC: -8935.38 The Akaike Information Criterion (AIC) is a measure of model complexity and fit. A lower AIC indicates a better model.

    sigma^2: 0.0001553 This is the estimated variance of the residuals, representing the model’s ability to fit the data. A smaller value suggests a better fit.

    Implication: The ARIMA(4,0,4) model provides a parsimonious representation of the LRNSEI time series. The model suggests that the current value of the series is influenced by the previous period’s error.

  3. The significance test indicate that the MA1 coefficient in the ARIMA(4,0,4) model is not statistically significant. The intercept coefficient is close to being statistically significant, but more evidence is needed to confirm its significance.

3.N225

acf(LRN225) 

pacf(LRN225) 


#Using Auto arima 
arma_pq_LRN225 = auto.arima(LRN225); arma_pq_LRN225
Series: LRN225 
ARIMA(0,0,0) with zero mean 

sigma^2 = 0.0001486:  log likelihood = 4371.79
AIC=-8741.58   AICc=-8741.58   BIC=-8736.29
#Application of ARIMA

model2 = arima(LRN225,order = c(0,0,0))
model2

Call:
arima(x = LRN225, order = c(0, 0, 0))

Coefficients:
      intercept
          2e-04
s.e.      3e-04

sigma^2 estimated as 0.0001485:  log likelihood = 4372.08,  aic = -8740.15
et = residuals(model2)
et
Time Series:
Start = 1 
End = 1463 
Frequency = 1 
   [1]  8.576735e-03  5.454471e-03 -2.835607e-03 -3.516030e-03 -2.631811e-03  2.336673e-03  9.699757e-03 -3.732455e-03 -4.649030e-03  1.637493e-03  1.058558e-04
  [12]  1.260052e-02 -7.871604e-03 -1.163779e-02 -1.831628e-03 -3.489563e-04 -1.462188e-02 -8.591574e-03  1.640918e-02 -9.290982e-03 -2.602584e-02 -4.864933e-02
  [23]  1.382822e-03  1.103621e-02 -2.373219e-02 -6.713332e-03 -4.510876e-03  1.434432e-02  1.158079e-02  1.931544e-02 -1.041122e-02  1.841272e-03 -1.096611e-02
  [34]  6.925355e-03  1.160311e-02  1.036532e-02 -1.471009e-02 -1.594160e-02 -2.554593e-02 -6.851514e-03  1.745434e-02 -7.976979e-03  5.171421e-03  4.480122e-03
  [45]  1.615088e-02  6.338334e-03 -8.965168e-03  9.820440e-04 -6.103361e-03 -9.306414e-03 -4.904305e-03  9.579691e-03 -4.640619e-02  6.922745e-03  2.595665e-02
  [56] -1.374902e-02  5.815400e-03  1.361457e-02 -3.309415e-03 -4.753573e-03  1.038112e-03  1.492793e-02 -3.846849e-03  4.880021e-03  5.098070e-03 -5.173230e-03
  [67] -1.478885e-03  5.212714e-03  2.362726e-03  3.107578e-04  1.387556e-02  1.245877e-03 -1.546374e-03 -3.595133e-03  8.327324e-03 -3.064263e-03  4.442056e-03
  [78]  6.379189e-03  1.544427e-03 -1.808761e-03 -4.914989e-04  1.605313e-03 -4.685511e-03  3.691181e-03  1.130646e-02  4.465672e-03 -2.335813e-03 -4.668292e-03
  [89]  5.076865e-03  3.778365e-03  2.894022e-03 -2.070269e-03 -1.209695e-02 -1.144246e-02  3.725184e-04  1.107318e-03 -5.712513e-03 -1.556095e-02  8.048960e-03
 [100] -1.614810e-03  1.340306e-02  2.584254e-03  3.575291e-03  8.450981e-03 -5.899002e-03  4.573642e-03  3.011934e-03  3.598988e-03 -1.020854e-02  4.721942e-03
 [111] -7.771094e-03 -1.811825e-02  1.211318e-02  5.840981e-03 -8.036621e-03 -8.208462e-03 -6.910703e-05 -3.389805e-03 -3.033402e-04  1.289443e-03 -2.257326e-02
 [122] -1.452075e-03 -3.390667e-03 -8.102446e-03  1.088825e-02  1.180421e-02  6.299343e-03 -1.223727e-02  1.135204e-02  1.804136e-02 -2.414256e-04  4.174540e-03
 [133]  4.015639e-03 -1.536882e-03 -3.180058e-03 -1.358641e-02  4.812988e-03  4.357814e-03 -1.452938e-03  5.316289e-03 -7.661658e-03  1.524167e-04  8.278570e-03
 [144] -1.058946e-02  3.203433e-04 -1.034603e-03  6.640147e-03 -1.054972e-03 -2.271364e-03 -1.361954e-02 -2.020111e-02  2.231600e-02 -7.057356e-03 -7.901944e-04
 [155]  3.282534e-03 -3.451766e-03  6.919856e-04  6.165640e-03  1.914747e-03  8.242884e-03  8.475147e-03  3.649825e-04  1.280638e-03  6.894738e-04 -4.316363e-04
 [166] -7.165340e-03 -7.030767e-04 -5.368474e-03 -4.363604e-03 -8.317322e-03  2.714220e-03  1.270787e-02 -2.895768e-03  9.299935e-03  1.166523e-02  1.377010e-02
 [177]  1.046006e-02 -1.396184e-04  7.961400e-03  2.700630e-03  3.657741e-03 -1.015354e-02  1.325296e-02  4.957330e-03  7.833573e-04 -6.841408e-03 -5.870529e-03
 [188] -8.277528e-03 -1.354572e-02  1.318898e-03 -3.995342e-02  4.342864e-03 -1.907220e-02  1.216106e-02  1.261958e-02 -8.283749e-03 -5.821409e-03  3.423957e-03
 [199] -2.731458e-02  3.404693e-03 -3.818190e-02 -4.204882e-03 -1.885424e-03  1.419249e-02  2.111466e-02 -1.091892e-02  2.507261e-02 -1.585793e-02  1.105396e-02
 [210] -3.042433e-03  1.775747e-02 -1.082198e-02  6.404613e-04 -2.108416e-02  1.406005e-03 -2.205286e-03 -5.911545e-03  6.232871e-03 -1.121009e-02 -3.749386e-03
 [221]  6.201171e-03  7.372727e-03  6.174788e-03  9.938680e-03  3.610101e-03  3.724223e-03  9.717260e-03 -2.439410e-02 -5.552310e-03 -1.948208e-02  7.959608e-03
 [232] -2.165013e-02 -3.615734e-03  2.103293e-02  9.590283e-03 -2.067969e-02  5.917433e-03 -1.860939e-02 -6.299358e-03 -2.901734e-02 -1.140513e-02 -5.164642e-02
 [243]  8.662367e-03  3.785808e-02 -3.376667e-03 -2.312499e-02  2.385058e-02  7.962200e-03  1.073656e-02 -1.321299e-02  9.427037e-03  9.319439e-03 -5.731412e-03
 [254] -2.223576e-03  1.260567e-02  2.332419e-03 -4.905909e-03 -1.657818e-03 -1.168829e-03  9.380820e-03 -6.255584e-03  5.157396e-04 -5.486390e-03  1.025715e-02
 [265]  4.755964e-04  4.336166e-03 -2.126017e-03  1.178152e-03 -6.140794e-03 -2.059578e-02  2.554033e-02  1.310218e-02 -4.670196e-04 -1.161539e-02  1.783381e-02
 [276]  7.354925e-04  5.788425e-03  1.285078e-03 -2.047019e-03  4.541436e-03 -3.910309e-03  4.740187e-03 -8.222043e-03  9.879162e-03  9.861181e-03 -4.639296e-03
 [287] -6.218337e-03 -6.782288e-03 -2.050734e-02  4.481134e-03  1.752167e-02 -1.021719e-02 -3.927131e-04  7.425353e-03  5.969786e-03 -1.059495e-03  1.707368e-03
 [298]  6.106335e-04 -3.076786e-02  2.104347e-02 -2.561609e-03 -1.650920e-02  7.905046e-03  1.395616e-02 -4.143330e-04  9.379542e-03  2.990338e-04  3.551188e-03
 [309] -2.346109e-03  1.638073e-03 -5.530889e-03  8.558613e-04  7.063453e-03  1.331696e-02  2.126220e-03  2.289408e-03 -8.709344e-03  4.745702e-03  5.393239e-04
 [320]  1.639963e-03 -2.928813e-03  4.592820e-03 -2.433648e-03 -1.540655e-02 -1.499741e-02 -9.564143e-03 -2.918147e-03 -7.465451e-03 -6.112394e-03  5.501239e-03
 [331] -6.185845e-03  8.602678e-03  2.185763e-03 -1.616966e-03  2.717791e-04 -6.473567e-03 -1.846405e-03  2.848869e-03  3.413412e-03 -1.239255e-02 -3.142300e-03
 [342] -1.667460e-02 -9.522106e-03 -3.561649e-04  1.760845e-02 -3.406092e-04  5.071846e-03  1.164420e-02  3.058602e-03 -3.763827e-03 -4.876952e-03  3.786711e-03
 [353]  9.518546e-05 -7.429147e-03  1.683237e-02  5.786634e-03 -9.801966e-03  1.044265e-03 -4.581363e-03 -5.313323e-03  1.161877e-02 -3.162997e-03  2.087511e-02
 [364]  8.761669e-04 -5.593034e-03  2.725266e-03  1.780799e-03 -1.003946e-02  1.187862e-03 -1.711074e-03  4.856151e-03  1.714340e-03 -7.212596e-03 -3.314149e-03
 [375] -2.013790e-02  1.955306e-02 -2.582693e-03  9.242979e-03  3.852212e-03  1.920280e-03 -4.774478e-03 -2.152444e-03  4.028974e-03 -8.928855e-03  6.624202e-04
 [386] -2.153474e-02 -1.779239e-02 -6.777053e-03 -3.586775e-03  3.494372e-03  4.190498e-03 -1.139270e-02  9.473496e-03 -1.239334e-02  4.032936e-04  6.803146e-03
 [397]  5.290087e-03 -3.081930e-03  2.162828e-04  3.769347e-03 -2.220226e-02  9.338943e-03  8.988974e-04 -1.144702e-03  1.158612e-02 -4.315507e-03 -3.946157e-07
 [408]  9.205794e-04  2.069131e-02  5.133046e-03  5.349145e-03  3.208767e-03  9.326492e-03  7.224450e-03  1.021310e-02  3.510439e-04 -2.088901e-03  3.564418e-03
 [419]  1.328740e-03  6.526861e-04 -3.808576e-03  1.033404e-03 -7.951496e-03 -5.881925e-03  5.688802e-03 -5.125607e-03 -2.050490e-02  2.961191e-03 -1.875122e-03
 [430]  9.652245e-03 -6.346730e-03  4.204211e-03  1.114896e-02  1.831753e-02  1.165255e-02 -1.179017e-03  1.575050e-03  2.254967e-03  3.144596e-03  5.277753e-03
 [441]  1.919299e-03  2.712958e-03  4.420802e-03 -5.960323e-03  3.425593e-03 -3.573590e-03  1.716451e-02  1.985154e-03  8.950810e-04  2.393248e-03 -2.810970e-03
 [452]  7.791168e-03 -8.787214e-03 -7.917439e-03  6.724691e-03  4.614717e-03 -5.555544e-03 -6.446281e-03 -5.004241e-03  2.978444e-03  7.513252e-03  3.209036e-03
 [463]  2.512164e-03 -1.463658e-03 -5.176037e-03  9.821568e-03 -6.623526e-03 -1.075769e-02  6.859207e-03  2.086778e-03  3.020249e-03 -1.117146e-03 -1.024728e-03
 [474]  1.166243e-03  2.497869e-02 -3.190853e-03  4.497157e-03 -5.728437e-03 -3.152772e-03 -2.263962e-03 -5.340173e-05  1.560715e-04 -2.245521e-03  5.713627e-03
 [485] -3.892786e-03 -7.867700e-03 -1.952267e-02  1.561426e-02 -1.610140e-02  2.255703e-02  4.410827e-03  7.052472e-03 -4.771508e-03  4.503563e-04  4.266358e-03
 [496]  1.514427e-03 -9.374254e-03  6.723250e-03 -1.010669e-02  1.091564e-03 -2.074937e-02 -5.731172e-03  6.784639e-03 -1.757038e-02  9.607734e-03 -1.034349e-02
 [507]  4.650416e-03  9.885803e-03  2.323888e-02 -2.153706e-03 -6.218634e-03  7.129410e-03 -1.645545e-03 -6.140198e-03 -7.203823e-03 -1.434527e-02  8.639442e-03
 [518]  3.105481e-03 -4.185030e-03 -3.422137e-02 -8.201243e-03 -2.178435e-02 -3.762091e-02  9.225962e-03 -1.256160e-02  5.802403e-04  1.055590e-02 -2.778046e-02
 [529] -5.221995e-02  8.268957e-03 -2.320690e-02 -4.535317e-02 -6.297712e-02 -2.516129e-02  3.166008e-04 -1.713541e-02 -1.068166e-02  1.979173e-02  6.865751e-02
 [540]  7.707234e-02 -4.641570e-02  3.785797e-02 -1.606833e-02 -9.081072e-03 -4.630386e-02 -1.399086e-02 -1.590016e-04  4.131314e-02  1.968537e-02  2.080502e-02
 [551] -6.275183e-04  7.622347e-03 -2.385836e-02  3.054571e-02 -4.769282e-03 -1.362415e-02  3.074344e-02 -1.177360e-02 -2.018257e-02 -7.676900e-03  1.487475e-02
 [562] -8.896596e-03  2.645847e-02 -8.497648e-04  2.090293e-02 -2.909527e-02  2.579351e-03  2.506841e-02  1.018862e-02 -1.427951e-03 -5.135408e-03 -1.777574e-02
 [573]  5.900496e-03  4.551060e-03  1.453526e-02  7.640979e-03 -2.323686e-03 -8.260448e-03  1.694800e-02  2.496743e-02  6.695082e-03  2.270057e-02 -1.995993e-03
 [584]  8.156385e-03  1.161862e-02  1.258271e-02  3.377266e-03  7.133155e-03  1.341459e-02 -4.005076e-03  1.226462e-03 -2.884292e-02 -7.719607e-03 -3.558244e-02
 [595]  4.742965e-02 -5.856754e-03 -4.717938e-03  5.260102e-03 -2.090187e-03  4.728151e-03 -8.949024e-04 -1.249905e-02  1.102876e-02 -2.347655e-02  1.299637e-02
 [606] -7.735747e-03  8.532993e-04  6.980685e-03  1.788216e-02 -4.642576e-03 -8.056161e-03  3.789837e-03 -1.088311e-02  2.167509e-02 -8.957498e-03  1.550546e-02
 [617] -7.903602e-03 -3.493889e-03  6.860679e-04  7.071502e-03 -6.053162e-03 -1.814409e-03 -2.818662e-03 -1.179519e-02 -2.828984e-03 -2.881279e-02  2.186979e-02
 [628]  1.665812e-02 -2.850098e-03 -4.545584e-03 -4.183987e-03  1.840593e-02  3.869660e-03  1.736011e-02  1.466821e-03 -8.546088e-03 -2.220714e-03  2.337740e-03
 [639] -1.024299e-02  1.491363e-03  2.599567e-03  1.320927e-02 -4.951477e-04 -3.768332e-03 -1.439650e-02  1.093192e-02 -3.144391e-04  4.461809e-03  9.108520e-03
 [650] -1.138764e-02 -5.230291e-03  7.703631e-03 -1.067595e-02  8.530637e-03  7.091898e-03  6.265910e-03 -4.683077e-03  6.381179e-04 -6.915689e-03  1.512299e-03
 [661] -8.327977e-04 -1.138287e-02  4.804715e-03  1.290195e-02  9.266957e-04 -1.539362e-02 -6.958693e-03  1.193947e-02  4.960750e-03 -7.071089e-04  9.287031e-03
 [672] -1.399993e-03 -2.827357e-03  1.585946e-03  8.151893e-04 -5.312090e-03 -4.359249e-03  1.082454e-02 -4.648538e-03  2.826878e-03 -7.253907e-03  1.559790e-03
 [683] -1.188014e-03 -6.049435e-04 -3.110717e-03 -3.944937e-03 -1.556523e-02  1.351853e-02  1.677299e-02  1.691567e-02  8.841803e-03  2.069328e-02  2.402035e-03
 [694]  1.742924e-02  6.492613e-03 -5.545714e-03  2.007638e-02  3.906741e-03 -1.131472e-02 -3.893932e-03 -4.423102e-03  2.445258e-02  4.762908e-03  8.860746e-03
 [705]  3.797554e-03 -8.195440e-03  1.305874e-02  2.602269e-04  7.152414e-05 -2.412011e-03 -7.888961e-03 -3.273027e-03  1.292792e-02 -2.544746e-03 -4.125464e-03
 [716]  2.752675e-03 -1.911189e-03  2.361635e-03  1.598223e-03 -1.857228e-03 -2.072866e-03 -1.070342e-02  3.096813e-03  5.156297e-03 -6.817535e-04  7.137325e-03
 [727]  2.600383e-02 -4.748804e-03 -7.034154e-03 -3.907563e-03 -4.029762e-03  1.567908e-02  2.308907e-02  6.576520e-04  1.008170e-02  8.215297e-03 -6.501079e-03
 [738] -1.000056e-02  1.351687e-02 -4.097536e-03  7.915014e-03 -4.612015e-03  6.401851e-03 -9.867323e-03  2.872571e-03 -1.564806e-02 -1.936201e-02  1.509972e-02
 [749]  9.363739e-03  9.733634e-03 -1.092969e-02  1.506810e-02  2.070948e-02  3.746383e-03  1.688526e-03 -1.692245e-03  1.868666e-02  1.242885e-02 -6.020267e-03
 [760] -2.095092e-03 -7.483130e-03  4.348920e-03 -1.643260e-02  1.635557e-02 -4.090905e-02  2.355284e-02 -8.886235e-03  4.877686e-03 -2.175016e-02 -2.518084e-03
 [771] -4.444710e-03  9.614413e-03  5.552329e-05  5.770111e-03  1.693850e-02  1.410783e-03  4.922739e-03 -4.673709e-04  9.817348e-03 -1.439623e-02 -2.119997e-02
 [782] -6.369354e-03 -2.081308e-02  1.111282e-02  1.519127e-02  6.856145e-03  1.396860e-03 -8.905255e-03  6.932131e-03  1.546141e-02  7.607705e-03 -1.337579e-02
 [793]  9.081517e-04 -9.752304e-04  1.745227e-03 -7.975151e-03  6.939502e-03 -4.641413e-03  4.908685e-04  1.129968e-03 -1.740501e-04 -2.014447e-02 -2.078862e-02
 [804]  2.331799e-02 -5.997927e-03  3.390747e-03 -4.864427e-03  1.897576e-03 -8.582722e-03  1.760227e-02  6.599729e-04  5.211373e-03 -3.154617e-02 -1.648955e-02
 [815] -2.540665e-02  2.268169e-02 -9.529415e-03  2.045974e-02 -1.308066e-02  1.675148e-03  7.542922e-03  1.409149e-03  6.412702e-03  2.843014e-03 -3.499961e-03
 [826]  2.057100e-02 -1.021678e-02 -1.827579e-03  4.322283e-03  3.619288e-03 -4.261796e-03  2.440415e-03 -2.161985e-03 -3.795633e-03  3.140136e-03 -5.809365e-04
 [837]  7.091884e-03  9.297391e-03 -5.359268e-03 -9.594359e-03 -2.112683e-03 -3.370309e-02  3.045613e-02 -5.613837e-04 -2.296561e-04  6.349712e-03 -8.664073e-04
 [848] -8.378615e-03 -9.733202e-04 -3.180292e-03  2.410857e-03 -6.692654e-03  1.331614e-03 -9.933168e-03 -9.055151e-03 -6.578025e-03  2.200709e-02  4.968146e-03
 [859] -4.070360e-03 -1.182230e-02 -1.004958e-02 -1.283111e-02 -9.855444e-03  5.577703e-03  1.006139e-02  4.666220e-03 -1.423076e-02  7.010953e-03 -1.835945e-02
 [870]  1.782619e-02 -5.264254e-03 -2.332836e-03  4.966807e-03  3.068135e-03  2.203865e-03  6.276244e-03 -2.220198e-03 -1.594084e-03 -1.660064e-02 -3.834622e-03
 [881]  5.628009e-03 -1.134985e-02 -1.011066e-02  1.740762e-02  8.372611e-03 -5.046508e-04  3.891567e-04 -3.894090e-03  5.103970e-03  1.050515e-02  1.254534e-02
 [892]  3.004160e-03  2.003266e-02  1.785056e-02  8.361080e-03  8.579996e-03 -5.990676e-03  1.213330e-02  1.913109e-03  7.047209e-03 -5.419076e-03 -6.434293e-03
 [903]  5.569215e-03 -2.212969e-02 -6.976943e-03  2.011087e-02 -5.307350e-04 -2.098291e-03 -2.166165e-02 -3.347653e-03 -2.365528e-02 -1.164325e-02 -2.237862e-02
 [914] -1.083754e-02  5.168828e-03  1.306390e-02  1.564865e-02 -9.675497e-03 -3.446277e-03  1.424610e-02  1.772869e-02 -1.727633e-03  6.285225e-03  1.127842e-03
 [925] -1.911472e-02  3.106301e-03 -7.364129e-03  1.728219e-02 -5.084003e-04 -9.846417e-03  2.274469e-03  2.553361e-02 -4.506566e-03  8.979495e-03 -6.395672e-03
 [936] -3.777355e-03 -7.779491e-03 -6.361492e-03  5.619039e-03  1.103815e-02  5.377015e-03  8.097922e-04 -4.268194e-03 -3.266372e-03  4.719749e-03  7.075074e-04
 [947] -1.620233e-02  6.446105e-03 -2.591317e-02 -1.664207e-02 -1.671643e-02  3.842681e-03 -6.786729e-03  9.661367e-03 -3.894280e-03  1.849636e-02  1.389162e-02
 [958] -4.935222e-03 -1.030746e-02  6.861859e-03 -7.525082e-03  7.105504e-04  2.084892e-02 -1.831595e-02 -2.176604e-02  2.029867e-02  1.322038e-03  7.992785e-03
 [969] -7.894990e-04 -3.935501e-03  1.335979e-02 -5.839591e-03 -4.233553e-03  1.731962e-02  7.945304e-04 -2.944760e-02 -5.682354e-04 -9.274128e-03  1.882205e-02
 [980] -9.901104e-03 -1.313083e-02  7.170879e-03 -2.936903e-03 -2.859781e-02  1.082669e-02 -9.308096e-03  2.157727e-03 -1.694619e-02 -4.674527e-03 -3.187263e-02
 [991]  2.044615e-02  1.035600e-02  2.587694e-03  1.642630e-02 -1.091390e-02  7.025430e-03 -7.230852e-03  1.066045e-03  1.052521e-02  3.963337e-03
 [ reached getOption("max.print") -- omitted 463 entries ]
#Checking of significance of ARMA
coeftest(model2)

z test of coefficients:

            Estimate Std. Error z value Pr(>|z|)
intercept 0.00024143 0.00031970  0.7552   0.4501
#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))

Explanation:

  1. Auto ARIMA: gave order of ARIMA (0,0,0)

    i.e. no autoregressive (AR) terms, no differencing (D) terms, and one moving average (MA) term.

  2. Log likelihood: 4371.79 This is a measure of the model’s goodness of fit. A higher log likelihood indicates a better fit.

    AIC: -8741.58 The Akaike Information Criterion (AIC) is a measure of model complexity and fit. A lower AIC indicates a better model.

    sigma^2: 0.0001486 This is the estimated variance of the residuals, representing the model’s ability to fit the data. A smaller value suggests a better fit.

    Implication: The ARIMA(0,0,0) model provides a parsimonious representation of the LRNSEI time series. The model suggests that the current value of the series is influenced by the previous period’s error.

  3. The significance test indicate that the MA1 coefficient in the ARIMA(0,0,0) model is not statistically significant. The intercept coefficient is close to being statistically significant, but more evidence is needed to confirm its significance. 4.HSI

acf(LRHSI) 

pacf(LRHSI) 


#Using Auto arima 
arma_pq_LRHSI = auto.arima(LRHSI); arma_pq_LRHSI
Series: LRHSI 
ARIMA(0,0,0) with zero mean 

sigma^2 = 0.000208:  log likelihood = 4159.46
AIC=-8316.92   AICc=-8316.92   BIC=-8311.62
#Application of ARIMA

model2 = arima(LRHSI,order = c(0,0,0))
model2

Call:
arima(x = LRHSI, order = c(0, 0, 0))

Coefficients:
      intercept
         -4e-04
s.e.      4e-04

sigma^2 estimated as 0.0002079:  log likelihood = 4160.01,  aic = -8316.03
et = residuals(model2)
et
Time Series:
Start = 1 
End = 1475 
Frequency = 1 
   [1]  1.889215e-03  6.121947e-03  2.934417e-03  3.145762e-03  4.008989e-03  2.401995e-03  1.895516e-03  9.738628e-03 -1.953261e-03  1.829048e-02  2.857167e-03
  [12]  4.716666e-03  4.525143e-03  4.680066e-03  1.684505e-02  1.244405e-03 -8.879152e-03  1.558064e-02 -5.268549e-03 -1.057316e-02  8.944517e-03 -7.088350e-03
  [23] -8.409554e-04 -1.060230e-02 -5.212478e-02 -8.542519e-03  4.609348e-03 -3.109120e-02 -1.226141e-03  1.320785e-02  2.279875e-02  1.986055e-02 -7.406645e-03
  [34]  1.831529e-02 -1.454879e-02  1.008391e-02  7.769158e-03 -6.932032e-03 -1.325598e-02  6.842731e-03 -1.455989e-02 -2.266104e-02  2.106996e-02 -9.943777e-03
  [45]  1.543491e-02  1.147959e-02  1.950744e-02  6.200375e-04 -4.886015e-03  3.763946e-03 -8.466059e-04  7.688957e-04  1.541825e-03 -3.906436e-03 -1.059892e-02
  [56] -2.442757e-02  8.264905e-03  8.287246e-03 -2.487411e-02  2.751899e-03  3.272244e-03 -2.176442e-02  1.138642e-02  1.320036e-02  1.677220e-02  5.878454e-03
  [67] -1.757632e-03 -3.482443e-04 -1.572993e-02 -7.980500e-03  7.735645e-03  1.430448e-02 -9.097395e-03 -5.009015e-03  1.293672e-02 -9.712559e-03 -1.022827e-02
  [78]  9.450947e-03  1.767416e-02 -2.354014e-03 -1.305666e-02 -1.244973e-02  2.656380e-03  1.392377e-02  4.770597e-03  9.297831e-03  1.049762e-02  1.376865e-02
  [89] -1.201670e-02 -9.489414e-04 -5.021637e-03  3.806889e-03  6.381671e-03 -1.798088e-02  3.480382e-03 -5.224693e-03  7.049033e-03 -9.647634e-03 -1.373769e-02
 [100]  1.400152e-02  1.193586e-03  1.682259e-02  3.469838e-03  5.708093e-03  8.472657e-03 -1.735545e-02  3.796382e-03  1.661048e-03 -1.182994e-02 -8.923702e-03
 [111] -3.907509e-03 -2.775611e-02  8.102779e-03 -1.317062e-02  1.849449e-03 -1.254912e-02 -2.371049e-03 -1.795526e-02  5.356983e-03  1.633135e-02 -1.385018e-02
 [122] -1.030849e-02 -1.717147e-03  5.121634e-03  1.347751e-02  1.768521e-04 -1.260896e-02  6.351191e-03  1.959803e-03  8.931360e-04 -1.222786e-02 -1.888067e-03
 [133] -3.402307e-03  7.992166e-03  1.515071e-03  1.467679e-02  9.367162e-03 -4.449464e-03  1.198363e-03 -2.078387e-03 -4.843632e-03 -8.117390e-03 -2.194769e-02
 [144] -9.860082e-04  5.556938e-03  1.570918e-02  4.290291e-03  9.107292e-03 -8.054153e-03 -1.488174e-02 -6.200452e-03 -1.519625e-02 -7.819731e-03  4.568643e-03
 [155]  1.442887e-02  5.987061e-03  6.673125e-03 -4.527171e-03 -3.881754e-03  2.182447e-02  3.232796e-03  2.678424e-03 -8.526729e-03 -9.435398e-03 -5.936521e-03
 [166]  9.761657e-03 -2.602938e-02 -9.529210e-03  3.446997e-04 -1.304349e-02 -6.803022e-03 -2.543122e-03  2.548814e-02  1.041012e-02 -1.264733e-02  6.015538e-03
 [177]  1.223914e-02  2.956481e-03  1.756634e-02 -1.598670e-02  1.187353e-02 -3.249953e-03  3.019748e-03 -2.372152e-02 -9.008267e-04 -1.700828e-02 -1.533924e-03
 [188] -1.362725e-02 -7.378640e-04  1.164705e-03 -3.562592e-02  2.135276e-02 -1.351588e-02  1.070440e-03  9.192610e-05  4.583610e-03  2.328097e-02 -3.093227e-02
 [199] -3.430509e-03 -9.768444e-03 -1.074272e-02  4.206926e-03 -8.776218e-03  1.629978e-02  1.771056e-02  4.164533e-02 -2.066485e-02  7.562915e-03  1.417469e-03
 [210]  3.450798e-03 -2.375481e-02  1.614981e-03  6.605203e-03 -4.987077e-03  1.774175e-02  3.462024e-03  7.567014e-03 -1.997128e-02  5.456557e-03  2.238882e-03
 [221] -3.136959e-03  1.754494e-02 -1.283139e-03  1.362148e-02 -8.320383e-03  2.499078e-03  2.555174e-02  3.274853e-03 -1.590588e-02 -2.464803e-02 -3.152603e-03
 [232] -1.162402e-02  1.143473e-03  1.637098e-02  1.320583e-02 -1.593279e-02  1.337813e-04 -1.015329e-02  2.373874e-03 -9.000030e-03  5.451121e-03 -3.575293e-03
 [243] -6.352764e-03  1.387940e-03  1.369583e-02 -2.767322e-02 -2.234639e-03  2.255645e-02  8.543355e-03  1.932120e-03  2.282197e-02  2.625961e-03  5.878613e-03
 [254] -1.353678e-02  2.042073e-02  3.067631e-03 -5.064652e-03  1.284436e-02  4.289882e-03 -6.656323e-03  4.965593e-04  4.561854e-03  1.678594e-02  6.765836e-04
 [265] -1.248612e-03  4.424494e-03  1.117542e-02 -2.516285e-05  2.521690e-03 -1.174568e-03  7.437692e-03  1.371033e-03  1.190945e-02 -1.907726e-03 -1.846553e-02
 [276]  1.625949e-02 -3.807787e-03  1.047268e-02  4.450080e-03  6.883637e-03  5.344930e-03 -6.091893e-03 -1.135659e-04 -3.935590e-03  6.626423e-03  5.498273e-03
 [287]  4.641297e-04  3.015460e-03 -8.535232e-03 -1.893756e-02  1.008536e-02  1.493829e-02 -3.534710e-03  1.918920e-03  5.955031e-03  1.397732e-02  2.340186e-03
 [298] -4.548818e-03 -8.147855e-03  1.831492e-03 -2.007938e-02  1.920757e-03  6.026632e-03  2.028059e-03  9.945725e-03  1.781987e-02  2.511777e-03  1.253091e-02
 [309] -1.276430e-03  5.088024e-03  3.062298e-03 -8.637787e-04 -8.948762e-03  2.748257e-03 -2.922024e-03  1.104366e-02  2.224821e-04 -4.978069e-03  3.940813e-04
 [320] -4.872557e-03 -8.232279e-03  2.261327e-03  1.006914e-02 -6.106210e-03  8.612647e-03  4.971815e-03 -2.901236e-02  5.625812e-03 -1.193517e-02 -2.375850e-02
 [331]  8.807178e-03 -1.471773e-02  5.597423e-03  6.196695e-04 -1.129521e-02 -5.305622e-03 -4.307936e-03  2.153997e-03 -1.557010e-02  3.572954e-03 -2.015129e-03
 [342]  4.151971e-03 -5.283828e-03 -4.051595e-03 -7.521153e-03  1.259172e-04 -4.538234e-03  5.386451e-03  2.988084e-03  2.288623e-02  8.005639e-03 -1.706118e-02
 [353] -1.089006e-04 -6.087606e-03  4.399137e-03  1.032098e-02  2.565136e-02  1.266884e-02 -2.296015e-03  1.773618e-03 -1.114068e-02  1.671149e-03  1.444900e-02
 [364] -2.362274e-03  1.199194e-02 -3.126867e-04 -1.664943e-03 -3.327025e-04 -1.512537e-02 -7.237471e-03  3.534247e-03  8.414741e-03  1.794240e-03  3.314841e-03
 [375]  2.659320e-03 -5.298581e-04 -4.215218e-03  1.101010e-02 -1.340190e-02  3.745352e-03  2.414670e-03  2.854950e-03 -6.503119e-03 -9.917170e-03  1.820076e-03
 [386] -1.279294e-02 -7.268389e-03 -2.336073e-02 -2.852235e-02 -6.322663e-03  1.194724e-03  5.143225e-03 -6.576809e-03 -4.032290e-03 -2.087244e-02  1.224193e-03
 [397]  8.000685e-03  9.715950e-03  2.183170e-02 -1.901418e-03  1.861356e-03 -8.065700e-03  5.396247e-03 -1.885011e-02 -2.386277e-04 -1.500364e-03  3.825028e-03
 [408]  1.220368e-03 -3.429117e-03 -3.464223e-03  3.864571e-02  1.043353e-04  6.981590e-03  4.401191e-05  4.801559e-04  1.805331e-02 -2.238782e-03  1.013239e-02
 [419] -7.980875e-03 -1.200687e-02 -9.544696e-04 -1.032140e-02 -8.633536e-04 -7.705476e-03  2.626956e-03 -1.245912e-02  4.110265e-03 -2.956216e-03  5.676847e-03
 [430] -1.507257e-03  2.987916e-03 -1.074629e-02  3.193609e-03 -7.771474e-03  1.372308e-03  2.348502e-02  8.473861e-03 -2.811616e-04  6.426537e-03  7.279507e-03
 [441] -4.418224e-03  6.229895e-04  2.656644e-03 -7.832369e-03  9.060397e-03 -4.489136e-03  8.754557e-03 -3.498858e-03 -4.059488e-03  9.317323e-03  7.580401e-03
 [452]  1.673753e-02  5.323147e-03  5.840068e-04  6.106007e-03 -6.671804e-03 -2.615943e-02  5.533615e-03 -1.801922e-02 -8.973736e-03  5.075796e-04  1.376768e-02
 [463]  1.574461e-02 -7.170272e-03 -1.545104e-02  5.226800e-03  1.524751e-02 -2.540665e-03  1.882821e-03 -1.843785e-03 -2.016339e-02  4.116208e-03 -1.627370e-03
 [474] -1.213988e-02  6.304456e-03  1.106838e-02  2.574069e-04 -1.800992e-03  8.262237e-03  1.339692e-02  2.576533e-02 -6.115559e-03  1.252172e-02  1.848224e-03
 [485] -2.612224e-03  2.940344e-03  1.651885e-03 -1.118581e-03  1.327462e-02  3.718497e-03 -4.193581e-03  1.286622e-02 -2.834307e-03 -7.555899e-03  3.785508e-03
 [496] -7.906705e-03  1.709730e-02  3.094046e-03  1.139408e-02 -2.018779e-03 -3.474617e-03  4.191325e-03  6.379648e-03 -8.611353e-03 -2.815819e-02  1.302518e-02
 [507] -1.496268e-02  1.845589e-03 -2.824108e-02 -2.615050e-02 -4.779480e-03  2.078802e-03  1.242513e-02  4.538178e-03  2.644459e-02 -2.863302e-03 -5.568424e-03
 [518]  1.289064e-02  9.049901e-03 -2.977151e-03  3.476874e-03  5.558330e-03 -1.508231e-02  4.947030e-03 -1.293512e-03 -1.054349e-02 -1.763393e-02  3.088612e-03
 [529] -6.947761e-03  3.466404e-03 -2.412765e-02  6.565872e-03  1.338038e-04 -1.995430e-03  2.099555e-02 -2.308569e-02 -4.283415e-02  1.435602e-02 -5.961958e-03
 [540] -3.685327e-02 -1.103066e-02 -4.077508e-02  9.035919e-03 -4.228097e-02 -2.609213e-02  4.964468e-02 -4.945422e-02  4.401606e-02  3.779621e-02 -7.064831e-03
 [551]  6.028778e-03 -1.285767e-02  1.871008e-02 -2.178221e-02  8.774721e-03 -1.494983e-03  2.223267e-02  2.140153e-02 -1.133906e-02  1.406622e-02  5.937716e-03
 [562] -1.154679e-02 -5.374149e-03  1.583535e-02 -1.657431e-03 -2.190166e-02  4.580735e-03  3.902560e-03 -5.712559e-03  1.905239e-02  1.250474e-02  3.142785e-03
 [573] -4.229077e-02  1.112972e-02  1.159426e-02 -6.124648e-03  1.074682e-02  1.562639e-02 -1.419702e-02 -2.305427e-03 -1.420916e-02 -9.603551e-04  6.147625e-03
 [584]  1.915911e-02  8.792105e-04 -4.532145e-03 -5.680721e-02  1.358049e-03  1.905945e-02 -3.173824e-03 -6.867183e-03 -7.037427e-03  3.342335e-02  1.143312e-02
 [595]  1.404019e-02  2.065715e-03  1.684347e-02  6.514329e-04  1.165027e-02  9.576327e-05 -2.260573e-02 -6.934694e-03 -2.142185e-02  2.396725e-02  6.019690e-03
 [606] -2.782774e-04  7.682708e-03 -4.998431e-03  1.642140e-02 -4.667161e-03 -8.994449e-03 -9.787728e-03  5.562572e-03  2.852900e-02  1.025394e-02  3.776123e-02
 [617] -1.350213e-02  6.287467e-03  3.489175e-03 -1.819543e-02  2.131021e-03 -1.108748e-02  5.395327e-04 -1.985839e-02  5.128294e-03 -8.487946e-04  2.318634e-02
 [628] -2.239890e-02  8.544886e-03 -2.192713e-02 -3.745335e-03  7.260441e-03  4.840571e-03 -6.563867e-03 -4.279772e-03 -5.199941e-03  2.017080e-02  6.624955e-03
 [639] -6.479100e-03 -1.573757e-02 -5.910438e-03  2.123047e-02  1.449059e-02 -1.342294e-04 -1.496030e-03  6.898969e-03  1.185076e-03 -7.062652e-03 -1.511557e-02
 [650]  1.331737e-02  1.767481e-02 -2.166483e-03  6.131944e-04 -7.902595e-03  5.952981e-03 -9.289693e-03  7.044446e-04 -2.179956e-03 -4.093422e-03 -1.216605e-02
 [661] -3.898613e-03  1.804473e-03 -5.936508e-03 -5.976045e-03  8.169608e-03  5.968974e-03  4.140931e-03  1.064542e-04 -1.528967e-02  5.090206e-03 -2.045962e-02
 [672] -9.416628e-03  1.476086e-03 -1.794400e-02 -2.855800e-03  1.069768e-02 -8.183489e-03  8.248550e-03  1.346920e-02  9.311152e-03  1.126958e-02 -1.649596e-03
 [683] -2.677716e-03  2.215330e-02  1.100787e-03 -2.043733e-02  9.798385e-03  6.749699e-03  1.505639e-03  7.891363e-03  1.674935e-03  5.732184e-03 -4.900009e-03
 [694] -2.772745e-03 -4.563191e-03 -1.928716e-02  1.491459e-02  1.981734e-02 -1.756353e-03  3.241591e-02  1.058073e-03  1.211744e-02  1.130170e-02 -2.441826e-03
 [705] -1.803879e-03 -8.385979e-05  8.952716e-03  1.660717e-03  5.273925e-03 -6.687107e-03  3.976295e-03  1.704202e-03  4.238399e-03  3.457219e-03  5.992110e-03
 [716]  3.195878e-03 -2.038848e-02  8.944889e-03 -9.272798e-04  7.751730e-03  4.405607e-03 -1.194338e-02 -7.266113e-03  7.904295e-03 -3.092098e-03  3.995858e-03
 [727] -4.004508e-03 -6.534630e-03  1.000224e-02  8.603182e-03 -6.366912e-03 -6.874266e-03 -6.755564e-03  8.928509e-03  2.043177e-03 -2.334986e-03  9.995578e-03
 [738]  2.193937e-02  3.484997e-03  9.230706e-03  6.818567e-03  1.928516e-03 -4.810895e-03  1.229170e-02  1.470263e-03  1.351335e-02 -1.061600e-03  9.605038e-03
 [749]  3.093141e-03  1.045495e-02  2.704392e-02  1.113866e-02 -7.644189e-04 -1.577085e-02  2.425755e-02 -2.539163e-02 -2.799422e-03 -2.542457e-02 -9.003103e-03
 [760]  2.170316e-02  1.263535e-02  2.401755e-03 -6.245406e-03  6.393832e-03  1.445475e-03  5.725709e-03  1.929917e-02  4.873880e-03  1.920972e-02  1.133680e-02
 [771] -1.548330e-02  2.010048e-03 -1.026403e-02  1.065890e-02 -2.991039e-02  1.230039e-02 -3.665871e-02  1.656270e-02 -1.179058e-02  2.700227e-02 -2.138086e-02
 [782] -4.353705e-03 -1.894895e-02  8.504494e-03  5.051029e-03  1.679806e-02 -2.183018e-02  3.661485e-03  7.098002e-03  6.162105e-04  1.311231e-02 -1.381116e-02
 [793] -3.254421e-03 -1.312728e-02 -2.014077e-02 -2.692582e-04  1.593029e-02  4.607624e-04  8.800154e-03 -6.598445e-03  1.994941e-02 -8.767741e-03  1.195010e-02
 [804] -1.032401e-02 -8.197185e-03  1.938905e-03  1.445746e-02 -3.338398e-03  6.508381e-03  5.093405e-03  1.410500e-03 -1.739768e-02  5.045362e-03  1.157892e-02
 [815] -3.944977e-03  4.677399e-06  4.869636e-03  8.340694e-03 -1.954036e-02 -1.247602e-02  7.408824e-03 -4.490221e-03  8.088357e-03 -5.419108e-04 -1.293431e-04
 [826] -2.016261e-02  8.119159e-03 -1.792112e-02  1.147722e-02  6.318429e-03  1.447264e-02 -4.637226e-03  6.811704e-04 -1.229291e-03  1.779128e-02  9.181434e-03
 [837] -1.417596e-03  7.797399e-04  1.334761e-03  1.118297e-02 -5.403941e-03 -1.098778e-02 -1.261324e-03 -4.139352e-03  1.898148e-04 -9.525305e-04  2.642564e-04
 [848]  3.980989e-03 -6.689473e-03 -6.672774e-03  4.667012e-03  8.856439e-03 -1.050669e-02 -5.916701e-03  1.815605e-02  2.661325e-03  1.434562e-02 -2.856325e-04
 [859] -9.017965e-03 -5.352239e-03 -1.772071e-02 -5.518777e-03 -2.118438e-03 -3.611457e-03 -2.890997e-02  7.419217e-03  6.617934e-03  1.655157e-02 -5.917269e-03
 [870]  7.881130e-03  6.950908e-04 -1.816262e-02 -8.026627e-03 -8.779353e-04  1.856718e-02 -1.420643e-02 -4.183054e-02 -4.274447e-02  1.572133e-02  3.289238e-02
 [881] -1.316002e-02  1.092311e-02 -1.168494e-03  9.202256e-03 -8.036102e-03 -5.707915e-04  4.359452e-03  1.257960e-02  2.442616e-03 -4.958617e-03 -4.375736e-03
 [892] -7.600207e-03 -1.638264e-02  5.088943e-03 -2.112402e-02 -1.820840e-02  1.079806e-02  2.472167e-02 -9.265221e-04 -1.049411e-02  8.783512e-05  5.562756e-03
 [903]  1.359838e-02  6.147269e-03  2.779314e-03 -6.854027e-03  1.044524e-02  7.630481e-03 -8.468976e-04 -2.285633e-02  1.926633e-02 -1.468059e-02 -1.174899e-02
 [914] -1.816780e-02 -1.430789e-02  1.059515e-02 -3.313027e-02  5.460833e-03  1.227365e-02 -1.269785e-02  1.081461e-03  1.236845e-02  7.030090e-03 -3.173951e-03
 [925] -2.179294e-02  3.210710e-03 -5.332693e-03  3.061142e-02  5.890116e-03  1.982165e-02 -1.402257e-02  1.504384e-02  3.500292e-03  1.514044e-02  1.383049e-02
 [936] -4.149174e-03  4.590790e-03  5.899000e-04 -3.199644e-03 -1.545824e-02 -2.458078e-03 -6.614124e-03 -8.428322e-03 -1.780235e-03 -2.594627e-03  8.372475e-03
 [947] -1.376559e-02 -3.906343e-03  2.386037e-03  7.743188e-03  1.041985e-02  3.557522e-03  2.876629e-03  1.303050e-02 -2.085581e-03 -1.256839e-02 -1.031618e-02
 [958] -3.550403e-03 -1.169179e-02  1.769761e-03  2.606547e-03 -2.662991e-02 -9.130319e-03 -1.553632e-02  8.187850e-03  5.874859e-03 -5.406026e-04 -1.731980e-02
 [969]  2.719704e-02  9.453309e-04  1.108835e-02 -1.034674e-02 -1.321237e-03 -1.299598e-02 -8.751293e-03  2.729259e-03 -1.172800e-02 -1.910070e-02  1.030249e-02
 [980]  6.081295e-03  4.339382e-03  1.692485e-03  2.837552e-03 -7.974245e-03  1.497396e-03  1.267879e-02 -4.872626e-03  1.042858e-03 -1.616902e-02  7.598287e-03
 [991]  1.845644e-02  1.111281e-02  7.975279e-05  2.794498e-02  1.525125e-03 -1.508413e-03 -6.407204e-03 -3.960674e-03  1.019531e-03  3.399596e-02
 [ reached getOption("max.print") -- omitted 475 entries ]
#Checking of significance of ARMA
coeftest(model2)

z test of coefficients:

             Estimate  Std. Error z value Pr(>|z|)
intercept -0.00039473  0.00037631  -1.049   0.2942
#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))

Explanation:

  1. Auto ARIMA: gave order of ARIMA (0,0,0)

    i.e. no autoregressive (AR) terms, no differencing (D) terms, and no moving average (MA) terms.

  2. sigma^2: 0.0002079. This is the estimated variance of the residuals, representing the model’s ability to fit the data.

    Log likelihood: 4160.01. A higher log likelihood indicates a better fit.

    AIC: -8316.03. A lower AIC indicates a better model.

    Implication: The ARIMA(0,0,0) model suggests that the LRHSI time series is a white noise process. This indicates that there are no predictable patterns or trends in the data.

  3. The significance test The significance test suggests that the intercept coefficient in the ARIMA(0,0,0) model is not statistically significant. 5.N100

acf(LRN100) 

pacf(LRN100) 


#Using Auto arima 
arma_pq_LRN100 = auto.arima(LRN100); arma_pq_LRN100
Series: LRN100 
ARIMA(0,0,0) with zero mean 

sigma^2 = 0.0001351:  log likelihood = 4662.72
AIC=-9323.44   AICc=-9323.44   BIC=-9318.11
#Application of ARIMA

model2 = arima(LRHSI,order = c(0,0,0))
model2

Call:
arima(x = LRHSI, order = c(0, 0, 0))

Coefficients:
      intercept
         -4e-04
s.e.      4e-04

sigma^2 estimated as 0.0002079:  log likelihood = 4160.01,  aic = -8316.03
et = residuals(model2)
et
Time Series:
Start = 1 
End = 1475 
Frequency = 1 
   [1]  1.889215e-03  6.121947e-03  2.934417e-03  3.145762e-03  4.008989e-03  2.401995e-03  1.895516e-03  9.738628e-03 -1.953261e-03  1.829048e-02  2.857167e-03
  [12]  4.716666e-03  4.525143e-03  4.680066e-03  1.684505e-02  1.244405e-03 -8.879152e-03  1.558064e-02 -5.268549e-03 -1.057316e-02  8.944517e-03 -7.088350e-03
  [23] -8.409554e-04 -1.060230e-02 -5.212478e-02 -8.542519e-03  4.609348e-03 -3.109120e-02 -1.226141e-03  1.320785e-02  2.279875e-02  1.986055e-02 -7.406645e-03
  [34]  1.831529e-02 -1.454879e-02  1.008391e-02  7.769158e-03 -6.932032e-03 -1.325598e-02  6.842731e-03 -1.455989e-02 -2.266104e-02  2.106996e-02 -9.943777e-03
  [45]  1.543491e-02  1.147959e-02  1.950744e-02  6.200375e-04 -4.886015e-03  3.763946e-03 -8.466059e-04  7.688957e-04  1.541825e-03 -3.906436e-03 -1.059892e-02
  [56] -2.442757e-02  8.264905e-03  8.287246e-03 -2.487411e-02  2.751899e-03  3.272244e-03 -2.176442e-02  1.138642e-02  1.320036e-02  1.677220e-02  5.878454e-03
  [67] -1.757632e-03 -3.482443e-04 -1.572993e-02 -7.980500e-03  7.735645e-03  1.430448e-02 -9.097395e-03 -5.009015e-03  1.293672e-02 -9.712559e-03 -1.022827e-02
  [78]  9.450947e-03  1.767416e-02 -2.354014e-03 -1.305666e-02 -1.244973e-02  2.656380e-03  1.392377e-02  4.770597e-03  9.297831e-03  1.049762e-02  1.376865e-02
  [89] -1.201670e-02 -9.489414e-04 -5.021637e-03  3.806889e-03  6.381671e-03 -1.798088e-02  3.480382e-03 -5.224693e-03  7.049033e-03 -9.647634e-03 -1.373769e-02
 [100]  1.400152e-02  1.193586e-03  1.682259e-02  3.469838e-03  5.708093e-03  8.472657e-03 -1.735545e-02  3.796382e-03  1.661048e-03 -1.182994e-02 -8.923702e-03
 [111] -3.907509e-03 -2.775611e-02  8.102779e-03 -1.317062e-02  1.849449e-03 -1.254912e-02 -2.371049e-03 -1.795526e-02  5.356983e-03  1.633135e-02 -1.385018e-02
 [122] -1.030849e-02 -1.717147e-03  5.121634e-03  1.347751e-02  1.768521e-04 -1.260896e-02  6.351191e-03  1.959803e-03  8.931360e-04 -1.222786e-02 -1.888067e-03
 [133] -3.402307e-03  7.992166e-03  1.515071e-03  1.467679e-02  9.367162e-03 -4.449464e-03  1.198363e-03 -2.078387e-03 -4.843632e-03 -8.117390e-03 -2.194769e-02
 [144] -9.860082e-04  5.556938e-03  1.570918e-02  4.290291e-03  9.107292e-03 -8.054153e-03 -1.488174e-02 -6.200452e-03 -1.519625e-02 -7.819731e-03  4.568643e-03
 [155]  1.442887e-02  5.987061e-03  6.673125e-03 -4.527171e-03 -3.881754e-03  2.182447e-02  3.232796e-03  2.678424e-03 -8.526729e-03 -9.435398e-03 -5.936521e-03
 [166]  9.761657e-03 -2.602938e-02 -9.529210e-03  3.446997e-04 -1.304349e-02 -6.803022e-03 -2.543122e-03  2.548814e-02  1.041012e-02 -1.264733e-02  6.015538e-03
 [177]  1.223914e-02  2.956481e-03  1.756634e-02 -1.598670e-02  1.187353e-02 -3.249953e-03  3.019748e-03 -2.372152e-02 -9.008267e-04 -1.700828e-02 -1.533924e-03
 [188] -1.362725e-02 -7.378640e-04  1.164705e-03 -3.562592e-02  2.135276e-02 -1.351588e-02  1.070440e-03  9.192610e-05  4.583610e-03  2.328097e-02 -3.093227e-02
 [199] -3.430509e-03 -9.768444e-03 -1.074272e-02  4.206926e-03 -8.776218e-03  1.629978e-02  1.771056e-02  4.164533e-02 -2.066485e-02  7.562915e-03  1.417469e-03
 [210]  3.450798e-03 -2.375481e-02  1.614981e-03  6.605203e-03 -4.987077e-03  1.774175e-02  3.462024e-03  7.567014e-03 -1.997128e-02  5.456557e-03  2.238882e-03
 [221] -3.136959e-03  1.754494e-02 -1.283139e-03  1.362148e-02 -8.320383e-03  2.499078e-03  2.555174e-02  3.274853e-03 -1.590588e-02 -2.464803e-02 -3.152603e-03
 [232] -1.162402e-02  1.143473e-03  1.637098e-02  1.320583e-02 -1.593279e-02  1.337813e-04 -1.015329e-02  2.373874e-03 -9.000030e-03  5.451121e-03 -3.575293e-03
 [243] -6.352764e-03  1.387940e-03  1.369583e-02 -2.767322e-02 -2.234639e-03  2.255645e-02  8.543355e-03  1.932120e-03  2.282197e-02  2.625961e-03  5.878613e-03
 [254] -1.353678e-02  2.042073e-02  3.067631e-03 -5.064652e-03  1.284436e-02  4.289882e-03 -6.656323e-03  4.965593e-04  4.561854e-03  1.678594e-02  6.765836e-04
 [265] -1.248612e-03  4.424494e-03  1.117542e-02 -2.516285e-05  2.521690e-03 -1.174568e-03  7.437692e-03  1.371033e-03  1.190945e-02 -1.907726e-03 -1.846553e-02
 [276]  1.625949e-02 -3.807787e-03  1.047268e-02  4.450080e-03  6.883637e-03  5.344930e-03 -6.091893e-03 -1.135659e-04 -3.935590e-03  6.626423e-03  5.498273e-03
 [287]  4.641297e-04  3.015460e-03 -8.535232e-03 -1.893756e-02  1.008536e-02  1.493829e-02 -3.534710e-03  1.918920e-03  5.955031e-03  1.397732e-02  2.340186e-03
 [298] -4.548818e-03 -8.147855e-03  1.831492e-03 -2.007938e-02  1.920757e-03  6.026632e-03  2.028059e-03  9.945725e-03  1.781987e-02  2.511777e-03  1.253091e-02
 [309] -1.276430e-03  5.088024e-03  3.062298e-03 -8.637787e-04 -8.948762e-03  2.748257e-03 -2.922024e-03  1.104366e-02  2.224821e-04 -4.978069e-03  3.940813e-04
 [320] -4.872557e-03 -8.232279e-03  2.261327e-03  1.006914e-02 -6.106210e-03  8.612647e-03  4.971815e-03 -2.901236e-02  5.625812e-03 -1.193517e-02 -2.375850e-02
 [331]  8.807178e-03 -1.471773e-02  5.597423e-03  6.196695e-04 -1.129521e-02 -5.305622e-03 -4.307936e-03  2.153997e-03 -1.557010e-02  3.572954e-03 -2.015129e-03
 [342]  4.151971e-03 -5.283828e-03 -4.051595e-03 -7.521153e-03  1.259172e-04 -4.538234e-03  5.386451e-03  2.988084e-03  2.288623e-02  8.005639e-03 -1.706118e-02
 [353] -1.089006e-04 -6.087606e-03  4.399137e-03  1.032098e-02  2.565136e-02  1.266884e-02 -2.296015e-03  1.773618e-03 -1.114068e-02  1.671149e-03  1.444900e-02
 [364] -2.362274e-03  1.199194e-02 -3.126867e-04 -1.664943e-03 -3.327025e-04 -1.512537e-02 -7.237471e-03  3.534247e-03  8.414741e-03  1.794240e-03  3.314841e-03
 [375]  2.659320e-03 -5.298581e-04 -4.215218e-03  1.101010e-02 -1.340190e-02  3.745352e-03  2.414670e-03  2.854950e-03 -6.503119e-03 -9.917170e-03  1.820076e-03
 [386] -1.279294e-02 -7.268389e-03 -2.336073e-02 -2.852235e-02 -6.322663e-03  1.194724e-03  5.143225e-03 -6.576809e-03 -4.032290e-03 -2.087244e-02  1.224193e-03
 [397]  8.000685e-03  9.715950e-03  2.183170e-02 -1.901418e-03  1.861356e-03 -8.065700e-03  5.396247e-03 -1.885011e-02 -2.386277e-04 -1.500364e-03  3.825028e-03
 [408]  1.220368e-03 -3.429117e-03 -3.464223e-03  3.864571e-02  1.043353e-04  6.981590e-03  4.401191e-05  4.801559e-04  1.805331e-02 -2.238782e-03  1.013239e-02
 [419] -7.980875e-03 -1.200687e-02 -9.544696e-04 -1.032140e-02 -8.633536e-04 -7.705476e-03  2.626956e-03 -1.245912e-02  4.110265e-03 -2.956216e-03  5.676847e-03
 [430] -1.507257e-03  2.987916e-03 -1.074629e-02  3.193609e-03 -7.771474e-03  1.372308e-03  2.348502e-02  8.473861e-03 -2.811616e-04  6.426537e-03  7.279507e-03
 [441] -4.418224e-03  6.229895e-04  2.656644e-03 -7.832369e-03  9.060397e-03 -4.489136e-03  8.754557e-03 -3.498858e-03 -4.059488e-03  9.317323e-03  7.580401e-03
 [452]  1.673753e-02  5.323147e-03  5.840068e-04  6.106007e-03 -6.671804e-03 -2.615943e-02  5.533615e-03 -1.801922e-02 -8.973736e-03  5.075796e-04  1.376768e-02
 [463]  1.574461e-02 -7.170272e-03 -1.545104e-02  5.226800e-03  1.524751e-02 -2.540665e-03  1.882821e-03 -1.843785e-03 -2.016339e-02  4.116208e-03 -1.627370e-03
 [474] -1.213988e-02  6.304456e-03  1.106838e-02  2.574069e-04 -1.800992e-03  8.262237e-03  1.339692e-02  2.576533e-02 -6.115559e-03  1.252172e-02  1.848224e-03
 [485] -2.612224e-03  2.940344e-03  1.651885e-03 -1.118581e-03  1.327462e-02  3.718497e-03 -4.193581e-03  1.286622e-02 -2.834307e-03 -7.555899e-03  3.785508e-03
 [496] -7.906705e-03  1.709730e-02  3.094046e-03  1.139408e-02 -2.018779e-03 -3.474617e-03  4.191325e-03  6.379648e-03 -8.611353e-03 -2.815819e-02  1.302518e-02
 [507] -1.496268e-02  1.845589e-03 -2.824108e-02 -2.615050e-02 -4.779480e-03  2.078802e-03  1.242513e-02  4.538178e-03  2.644459e-02 -2.863302e-03 -5.568424e-03
 [518]  1.289064e-02  9.049901e-03 -2.977151e-03  3.476874e-03  5.558330e-03 -1.508231e-02  4.947030e-03 -1.293512e-03 -1.054349e-02 -1.763393e-02  3.088612e-03
 [529] -6.947761e-03  3.466404e-03 -2.412765e-02  6.565872e-03  1.338038e-04 -1.995430e-03  2.099555e-02 -2.308569e-02 -4.283415e-02  1.435602e-02 -5.961958e-03
 [540] -3.685327e-02 -1.103066e-02 -4.077508e-02  9.035919e-03 -4.228097e-02 -2.609213e-02  4.964468e-02 -4.945422e-02  4.401606e-02  3.779621e-02 -7.064831e-03
 [551]  6.028778e-03 -1.285767e-02  1.871008e-02 -2.178221e-02  8.774721e-03 -1.494983e-03  2.223267e-02  2.140153e-02 -1.133906e-02  1.406622e-02  5.937716e-03
 [562] -1.154679e-02 -5.374149e-03  1.583535e-02 -1.657431e-03 -2.190166e-02  4.580735e-03  3.902560e-03 -5.712559e-03  1.905239e-02  1.250474e-02  3.142785e-03
 [573] -4.229077e-02  1.112972e-02  1.159426e-02 -6.124648e-03  1.074682e-02  1.562639e-02 -1.419702e-02 -2.305427e-03 -1.420916e-02 -9.603551e-04  6.147625e-03
 [584]  1.915911e-02  8.792105e-04 -4.532145e-03 -5.680721e-02  1.358049e-03  1.905945e-02 -3.173824e-03 -6.867183e-03 -7.037427e-03  3.342335e-02  1.143312e-02
 [595]  1.404019e-02  2.065715e-03  1.684347e-02  6.514329e-04  1.165027e-02  9.576327e-05 -2.260573e-02 -6.934694e-03 -2.142185e-02  2.396725e-02  6.019690e-03
 [606] -2.782774e-04  7.682708e-03 -4.998431e-03  1.642140e-02 -4.667161e-03 -8.994449e-03 -9.787728e-03  5.562572e-03  2.852900e-02  1.025394e-02  3.776123e-02
 [617] -1.350213e-02  6.287467e-03  3.489175e-03 -1.819543e-02  2.131021e-03 -1.108748e-02  5.395327e-04 -1.985839e-02  5.128294e-03 -8.487946e-04  2.318634e-02
 [628] -2.239890e-02  8.544886e-03 -2.192713e-02 -3.745335e-03  7.260441e-03  4.840571e-03 -6.563867e-03 -4.279772e-03 -5.199941e-03  2.017080e-02  6.624955e-03
 [639] -6.479100e-03 -1.573757e-02 -5.910438e-03  2.123047e-02  1.449059e-02 -1.342294e-04 -1.496030e-03  6.898969e-03  1.185076e-03 -7.062652e-03 -1.511557e-02
 [650]  1.331737e-02  1.767481e-02 -2.166483e-03  6.131944e-04 -7.902595e-03  5.952981e-03 -9.289693e-03  7.044446e-04 -2.179956e-03 -4.093422e-03 -1.216605e-02
 [661] -3.898613e-03  1.804473e-03 -5.936508e-03 -5.976045e-03  8.169608e-03  5.968974e-03  4.140931e-03  1.064542e-04 -1.528967e-02  5.090206e-03 -2.045962e-02
 [672] -9.416628e-03  1.476086e-03 -1.794400e-02 -2.855800e-03  1.069768e-02 -8.183489e-03  8.248550e-03  1.346920e-02  9.311152e-03  1.126958e-02 -1.649596e-03
 [683] -2.677716e-03  2.215330e-02  1.100787e-03 -2.043733e-02  9.798385e-03  6.749699e-03  1.505639e-03  7.891363e-03  1.674935e-03  5.732184e-03 -4.900009e-03
 [694] -2.772745e-03 -4.563191e-03 -1.928716e-02  1.491459e-02  1.981734e-02 -1.756353e-03  3.241591e-02  1.058073e-03  1.211744e-02  1.130170e-02 -2.441826e-03
 [705] -1.803879e-03 -8.385979e-05  8.952716e-03  1.660717e-03  5.273925e-03 -6.687107e-03  3.976295e-03  1.704202e-03  4.238399e-03  3.457219e-03  5.992110e-03
 [716]  3.195878e-03 -2.038848e-02  8.944889e-03 -9.272798e-04  7.751730e-03  4.405607e-03 -1.194338e-02 -7.266113e-03  7.904295e-03 -3.092098e-03  3.995858e-03
 [727] -4.004508e-03 -6.534630e-03  1.000224e-02  8.603182e-03 -6.366912e-03 -6.874266e-03 -6.755564e-03  8.928509e-03  2.043177e-03 -2.334986e-03  9.995578e-03
 [738]  2.193937e-02  3.484997e-03  9.230706e-03  6.818567e-03  1.928516e-03 -4.810895e-03  1.229170e-02  1.470263e-03  1.351335e-02 -1.061600e-03  9.605038e-03
 [749]  3.093141e-03  1.045495e-02  2.704392e-02  1.113866e-02 -7.644189e-04 -1.577085e-02  2.425755e-02 -2.539163e-02 -2.799422e-03 -2.542457e-02 -9.003103e-03
 [760]  2.170316e-02  1.263535e-02  2.401755e-03 -6.245406e-03  6.393832e-03  1.445475e-03  5.725709e-03  1.929917e-02  4.873880e-03  1.920972e-02  1.133680e-02
 [771] -1.548330e-02  2.010048e-03 -1.026403e-02  1.065890e-02 -2.991039e-02  1.230039e-02 -3.665871e-02  1.656270e-02 -1.179058e-02  2.700227e-02 -2.138086e-02
 [782] -4.353705e-03 -1.894895e-02  8.504494e-03  5.051029e-03  1.679806e-02 -2.183018e-02  3.661485e-03  7.098002e-03  6.162105e-04  1.311231e-02 -1.381116e-02
 [793] -3.254421e-03 -1.312728e-02 -2.014077e-02 -2.692582e-04  1.593029e-02  4.607624e-04  8.800154e-03 -6.598445e-03  1.994941e-02 -8.767741e-03  1.195010e-02
 [804] -1.032401e-02 -8.197185e-03  1.938905e-03  1.445746e-02 -3.338398e-03  6.508381e-03  5.093405e-03  1.410500e-03 -1.739768e-02  5.045362e-03  1.157892e-02
 [815] -3.944977e-03  4.677399e-06  4.869636e-03  8.340694e-03 -1.954036e-02 -1.247602e-02  7.408824e-03 -4.490221e-03  8.088357e-03 -5.419108e-04 -1.293431e-04
 [826] -2.016261e-02  8.119159e-03 -1.792112e-02  1.147722e-02  6.318429e-03  1.447264e-02 -4.637226e-03  6.811704e-04 -1.229291e-03  1.779128e-02  9.181434e-03
 [837] -1.417596e-03  7.797399e-04  1.334761e-03  1.118297e-02 -5.403941e-03 -1.098778e-02 -1.261324e-03 -4.139352e-03  1.898148e-04 -9.525305e-04  2.642564e-04
 [848]  3.980989e-03 -6.689473e-03 -6.672774e-03  4.667012e-03  8.856439e-03 -1.050669e-02 -5.916701e-03  1.815605e-02  2.661325e-03  1.434562e-02 -2.856325e-04
 [859] -9.017965e-03 -5.352239e-03 -1.772071e-02 -5.518777e-03 -2.118438e-03 -3.611457e-03 -2.890997e-02  7.419217e-03  6.617934e-03  1.655157e-02 -5.917269e-03
 [870]  7.881130e-03  6.950908e-04 -1.816262e-02 -8.026627e-03 -8.779353e-04  1.856718e-02 -1.420643e-02 -4.183054e-02 -4.274447e-02  1.572133e-02  3.289238e-02
 [881] -1.316002e-02  1.092311e-02 -1.168494e-03  9.202256e-03 -8.036102e-03 -5.707915e-04  4.359452e-03  1.257960e-02  2.442616e-03 -4.958617e-03 -4.375736e-03
 [892] -7.600207e-03 -1.638264e-02  5.088943e-03 -2.112402e-02 -1.820840e-02  1.079806e-02  2.472167e-02 -9.265221e-04 -1.049411e-02  8.783512e-05  5.562756e-03
 [903]  1.359838e-02  6.147269e-03  2.779314e-03 -6.854027e-03  1.044524e-02  7.630481e-03 -8.468976e-04 -2.285633e-02  1.926633e-02 -1.468059e-02 -1.174899e-02
 [914] -1.816780e-02 -1.430789e-02  1.059515e-02 -3.313027e-02  5.460833e-03  1.227365e-02 -1.269785e-02  1.081461e-03  1.236845e-02  7.030090e-03 -3.173951e-03
 [925] -2.179294e-02  3.210710e-03 -5.332693e-03  3.061142e-02  5.890116e-03  1.982165e-02 -1.402257e-02  1.504384e-02  3.500292e-03  1.514044e-02  1.383049e-02
 [936] -4.149174e-03  4.590790e-03  5.899000e-04 -3.199644e-03 -1.545824e-02 -2.458078e-03 -6.614124e-03 -8.428322e-03 -1.780235e-03 -2.594627e-03  8.372475e-03
 [947] -1.376559e-02 -3.906343e-03  2.386037e-03  7.743188e-03  1.041985e-02  3.557522e-03  2.876629e-03  1.303050e-02 -2.085581e-03 -1.256839e-02 -1.031618e-02
 [958] -3.550403e-03 -1.169179e-02  1.769761e-03  2.606547e-03 -2.662991e-02 -9.130319e-03 -1.553632e-02  8.187850e-03  5.874859e-03 -5.406026e-04 -1.731980e-02
 [969]  2.719704e-02  9.453309e-04  1.108835e-02 -1.034674e-02 -1.321237e-03 -1.299598e-02 -8.751293e-03  2.729259e-03 -1.172800e-02 -1.910070e-02  1.030249e-02
 [980]  6.081295e-03  4.339382e-03  1.692485e-03  2.837552e-03 -7.974245e-03  1.497396e-03  1.267879e-02 -4.872626e-03  1.042858e-03 -1.616902e-02  7.598287e-03
 [991]  1.845644e-02  1.111281e-02  7.975279e-05  2.794498e-02  1.525125e-03 -1.508413e-03 -6.407204e-03 -3.960674e-03  1.019531e-03  3.399596e-02
 [ reached getOption("max.print") -- omitted 475 entries ]
#Checking of significance of ARMA
coeftest(model2)

z test of coefficients:

             Estimate  Std. Error z value Pr(>|z|)
intercept -0.00039473  0.00037631  -1.049   0.2942
#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))

Explanation:

  1. Auto ARIMA: gave order of ARIMA (0,0,0)

    no autoregressive (AR) terms, no differencing (D) terms, and no moving average (MA) terms.

  2. sigma^2: 0.0001351. This is the estimated variance of the residuals, representing the model’s ability to fit the data.

    Log likelihood: 4662.94. A higher log likelihood indicates a better fit.

    AIC: -9321.89. A lower AIC indicates a better model.

    Implication: The ARIMA(0,0,0) model suggests that the LRN100 time series is a white noise process. This indicates that there are no predictable patterns or trends in the data.

  3. The significance test The significance test suggests that the intercept coefficient in the ARIMA(0,0,0) model is not statistically significant. This supports the conclusion that the LRN100 series is a white noise process with a mean close to zero.

Linearity Test:

Used Test: BDS Test

H0: There is linearity in log returns of series

H1: There is no linearity in log returns of series

bds_result_BSESN <- bds.test(LRBSESN)

# Print the result
print(bds_result_BSESN)

     BDS Test 

data:  LRBSESN 

Embedding dimension =  2 3 

Epsilon for close points =  0.0059 0.0117 0.0176 0.0234 

Standard Normal = 
      [ 0.0059 ] [ 0.0117 ] [ 0.0176 ] [ 0.0234 ]
[ 2 ]     10.290    10.0096    11.8203    13.5999
[ 3 ]     13.116    12.4284    13.8365    15.6771

p-value = 
      [ 0.0059 ] [ 0.0117 ] [ 0.0176 ] [ 0.0234 ]
[ 2 ]          0          0          0          0
[ 3 ]          0          0          0          0
#BSESN data is non linear
bds_result_GSPC <- bds.test(LRGSPC)

# Print the result
print(bds_result_GSPC)

     BDS Test 

data:  LRGSPC 

Embedding dimension =  2 3 

Epsilon for close points =  0.0065 0.0131 0.0196 0.0261 

Standard Normal = 
      [ 0.0065 ] [ 0.0131 ] [ 0.0196 ] [ 0.0261 ]
[ 2 ]    10.2639    10.3889    11.3072    12.6351
[ 3 ]    16.1656    14.8205    14.7769    15.6305

p-value = 
      [ 0.0065 ] [ 0.0131 ] [ 0.0196 ] [ 0.0261 ]
[ 2 ]          0          0          0          0
[ 3 ]          0          0          0          0
#GSPC data is non linear
bds_result_N225 <- bds.test(LRN225)

# Print the result
print(bds_result_N225)

     BDS Test 

data:  LRN225 

Embedding dimension =  2 3 

Epsilon for close points =  0.0061 0.0122 0.0183 0.0244 

Standard Normal = 
      [ 0.0061 ] [ 0.0122 ] [ 0.0183 ] [ 0.0244 ]
[ 2 ]     5.7808     5.7682     5.7902     6.9286
[ 3 ]     8.3245     8.2855     8.2031     8.7682

p-value = 
      [ 0.0061 ] [ 0.0122 ] [ 0.0183 ] [ 0.0244 ]
[ 2 ]          0          0          0          0
[ 3 ]          0          0          0          0
#N225 data is non linear
bds_result_HSI <- bds.test(LRHSI)

# Print the result
print(bds_result_HSI)

     BDS Test 

data:  LRHSI 

Embedding dimension =  2 3 

Epsilon for close points =  0.0072 0.0144 0.0216 0.0288 

Standard Normal = 
      [ 0.0072 ] [ 0.0144 ] [ 0.0216 ] [ 0.0288 ]
[ 2 ]     0.4446     1.2127     2.7394     4.7064
[ 3 ]     1.3701     2.2145     3.7766     5.9738

p-value = 
      [ 0.0072 ] [ 0.0144 ] [ 0.0216 ] [ 0.0288 ]
[ 2 ]     0.6566     0.2252     0.0062          0
[ 3 ]     0.1707     0.0268     0.0002          0
#HSI data is non linear
bds_result_N100 <- bds.test(LRN100)

# Print the result
print(bds_result_N100)

     BDS Test 

data:  LRN100 

Embedding dimension =  2 3 

Epsilon for close points =  0.0058 0.0116 0.0174 0.0233 

Standard Normal = 
      [ 0.0058 ] [ 0.0116 ] [ 0.0174 ] [ 0.0233 ]
[ 2 ]     9.4712     9.1370     9.0274     8.9367
[ 3 ]    13.4175    12.0955    11.1530    10.8504

p-value = 
      [ 0.0058 ] [ 0.0116 ] [ 0.0174 ] [ 0.0233 ]
[ 2 ]          0          0          0          0
[ 3 ]          0          0          0          0
#N100 data is non linear

Explanation:

p-value for all 5 series is less than 0.05

hence we reject the null hypothesis and say that there is no linearity in log returns of any series

Implication: Since series is not linear, a SETAR model might be a more appropriate choice for modeling the log return series. The SETAR model can capture the non-linear relationships and threshold effects present in the data.

BSESN:

VARselect(LRBSESN)
$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
    10      7      7     10 

$criteria
                   1            2             3             4             5             6             7             8             9            10
AIC(n) -8.8883459138 -8.887293323 -8.8859605318 -8.8851478225 -8.8999061241 -8.9137036050 -8.9179104639 -8.9168047171 -8.9163464069 -8.9187745869
HQ(n)  -8.8856525346 -8.883253254 -8.8805737735 -8.8784143746 -8.8918259866 -8.9042767779 -8.9071369472 -8.9046845109 -8.9028795111 -8.9039610015
SC(n)  -8.8811246025 -8.876461356 -8.8715179092 -8.8670945443 -8.8782421903 -8.8884290155 -8.8890252187 -8.8843088163 -8.8802398504 -8.8790573748
FPE(n)  0.0001379877  0.000138133  0.0001383173  0.0001384297  0.0001364017  0.0001345327  0.0001339679  0.0001341161  0.0001341776  0.0001338522
#AIC lag is 10
selectSETAR(LRBSESN, m=2, thDelay = 1)
Using maximum autoregressive order for low regime: mL = 2 
Using maximum autoregressive order for high regime: mH = 2 
Searching on 1031 possible threshold values within regimes with sufficient ( 15% ) number of observations
Searching on  4124  combinations of thresholds ( 1031 ), thDelay ( 1 ), mL ( 2 ) and MM ( 2 ) 
Results of the grid search for 1 threshold

mod.setar1 = setar(LRBSESN, m=2, thDelay = 1, th= -0.007846886)
summary(mod.setar1)

Non linear autoregressive model

SETAR model ( 2 regimes)
Coefficients:
Low regime:
      const.L        phiL.1        phiL.2 
-0.0006002744 -0.1619868545 -0.0369244206 

High regime:
      const.H        phiH.1        phiH.2 
 0.0005889211 -0.0012648234  0.0148180386 

Threshold:
-Variable: Z(t) = + (0) X(t)+ (1)X(t-1)
-Value: -0.007847 (fixed)
Proportion of points in low regime: 15.61%   High regime: 84.39% 

Residuals:
        Min          1Q      Median          3Q         Max 
-0.13210561 -0.00522834  0.00039675  0.00587742  0.08109691 

Fit:
residuals variance = 0.000136,  AIC = -13120, MAPE = 117.3%

Coefficient(s):

           Estimate  Std. Error  t value Pr(>|t|)   
const.L -0.00060027  0.00121338  -0.4947 0.620876   
phiL.1  -0.16198685  0.05178108  -3.1283 0.001793 **
phiL.2  -0.03692442  0.05778069  -0.6390 0.522894   
const.H  0.00058892  0.00036295   1.6226 0.104891   
phiH.1  -0.00126482  0.03018484  -0.0419 0.966582   
phiH.2   0.01481804  0.04060796   0.3649 0.715235   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold
Variable: Z(t) = + (0) X(t) + (1) X(t-1)

Value: -0.007847 (fixed)

Explanation:

AIC order: 10
Hence Threshold value: -0.007846886

Implication: The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series’ behavior.

GSPC:

VARselect(LRGSPC)
$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     9      9      9      9 

$criteria
                   1             2             3             4             5             6             7             8             9            10
AIC(n) -8.6948779057 -8.7006300470 -8.6996339786 -8.7033708540 -8.7040914022 -8.7155444784 -8.7338966229 -8.7427322452 -8.7593777134 -8.7592208139
HQ(n)  -8.6922357127 -8.6966667576 -8.6943495926 -8.6967653716 -8.6961648233 -8.7062968030 -8.7233278510 -8.7308423769 -8.7461667485 -8.7446887526
SC(n)  -8.6877859348 -8.6899920907 -8.6854500369 -8.6856409269 -8.6828154897 -8.6907225804 -8.7055287395 -8.7108183764 -8.7239178591 -8.7202149742
FPE(n)  0.0001674413  0.0001664809  0.0001666468  0.0001660252  0.0001659056  0.0001640164  0.0001610338  0.0001596172  0.0001569823  0.0001570069
#AIC lag is 9
selectSETAR(LRGSPC, m=2, thDelay = 1)
Using maximum autoregressive order for low regime: mL = 2 
Using maximum autoregressive order for high regime: mH = 2 
Searching on 1054 possible threshold values within regimes with sufficient ( 15% ) number of observations
Searching on  4216  combinations of thresholds ( 1054 ), thDelay ( 1 ), mL ( 2 ) and MM ( 2 ) 
Results of the grid search for 1 threshold

mod.setar2 = setar(LRGSPC, m=2, thDelay = 1, th= -0.007264115)
summary(mod.setar2)

Non linear autoregressive model

SETAR model ( 2 regimes)
Coefficients:
Low regime:
     const.L       phiL.1       phiL.2 
 0.006076111 -0.283161982  0.295352834 

High regime:
      const.H        phiH.1        phiH.2 
-0.0001563904 -0.0431681899  0.1046109439 

Threshold:
-Variable: Z(t) = + (0) X(t)+ (1)X(t-1)
-Value: -0.007264 (fixed)
Proportion of points in low regime: 18.53%   High regime: 81.47% 

Residuals:
        Min          1Q      Median          3Q         Max 
-0.10699514 -0.00570294  0.00050195  0.00650953  0.08828058 

Fit:
residuals variance = 0.0001597,  AIC = -13171, MAPE = 138.9%

Coefficient(s):

           Estimate  Std. Error  t value  Pr(>|t|)    
const.L  0.00607611  0.00129062   4.7079 2.734e-06 ***
phiL.1  -0.28316198  0.04096171  -6.9128 7.000e-12 ***
phiL.2   0.29535283  0.05995260   4.9264 9.300e-07 ***
const.H -0.00015639  0.00040545  -0.3857   0.69976    
phiH.1  -0.04316819  0.03267940  -1.3210   0.18672    
phiH.2   0.10461094  0.04074319   2.5676   0.01034 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold
Variable: Z(t) = + (0) X(t) + (1) X(t-1)

Value: -0.007264 (fixed)

Explanation:

AIC order: 9
Hence Threshold value: -0.007264115

Implication: The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series’ behavior.

N225:

VARselect(LRN225)
$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     7      2      1      7 

$criteria
                   1             2             3             4             5            6             7            8             9            10
AIC(n) -8.8064220023 -8.8089861754 -8.8084980041 -8.8079878686 -8.8076022254 -8.807592237 -8.8095343233 -8.808234382 -8.8070165531 -8.8059014050
HQ(n)  -8.8037094870 -8.8049174024 -8.8030729735 -8.8012065803 -8.7994646795 -8.798098433 -8.7986842621 -8.796028063 -8.7934539766 -8.7909825708
SC(n)  -8.7991523731 -8.7980817315 -8.7939587456 -8.7898137954 -8.7857933376 -8.782148534 -8.7804558062 -8.775521051 -8.7706684067 -8.7659184440
FPE(n)  0.0001497682  0.0001493846  0.0001494576  0.0001495338  0.0001495915  0.000149593  0.0001493028  0.000149497  0.0001496792  0.0001498462
#AIC lag is 7
selectSETAR(LRN225, m=2, thDelay = 1)
Using maximum autoregressive order for low regime: mL = 2 
Using maximum autoregressive order for high regime: mH = 2 
Searching on 1021 possible threshold values within regimes with sufficient ( 15% ) number of observations
Searching on  4084  combinations of thresholds ( 1021 ), thDelay ( 1 ), mL ( 2 ) and MM ( 2 ) 
Results of the grid search for 1 threshold

mod.setar3 = setar(LRN225, m=2, thDelay = 1, th= -0.008663830)
summary(mod.setar3)

Non linear autoregressive model

SETAR model ( 2 regimes)
Coefficients:
Low regime:
     const.L       phiL.1       phiL.2 
0.0009539415 0.0858228839 0.1293747986 

High regime:
      const.H        phiH.1        phiH.2 
 0.0004778877 -0.0240467414  0.0231179134 

Threshold:
-Variable: Z(t) = + (0) X(t)+ (1)X(t-1)
-Value: -0.008664 (fixed)
Proportion of points in low regime: 18.69%   High regime: 81.31% 

Residuals:
        Min          1Q      Median          3Q         Max 
-0.05684686 -0.00633326  0.00035811  0.00661996  0.07802954 

Fit:
residuals variance = 0.0001473,  AIC = -12896, MAPE = 121.6%

Coefficient(s):

           Estimate  Std. Error  t value Pr(>|t|)  
const.L  0.00095394  0.00165025   0.5781  0.56331  
phiL.1   0.08582288  0.04878113   1.7593  0.07873 .
phiL.2   0.12937480  0.08534870   1.5158  0.12978  
const.H  0.00047789  0.00039173   1.2200  0.22268  
phiH.1  -0.02404674  0.03089351  -0.7784  0.43647  
phiH.2   0.02311791  0.03976339   0.5814  0.56107  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold
Variable: Z(t) = + (0) X(t) + (1) X(t-1)

Value: -0.008664 (fixed)

Explanation:

AIC order: 7
Hence Threshold value: -0.008663830

Implication: The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series’ behavior.

HSI:

VARselect(LRHSI)
$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     6      1      1      6 

$criteria
                   1             2             3             4             5             6             7             8            9            10
AIC(n) -8.4707563079 -8.4698769448 -8.4690724160 -8.4689447709 -8.4675810792 -8.4726825328 -8.4714138743 -8.4705520617 -8.470720092 -8.4694221308
HQ(n)  -8.4680629288 -8.4658368760 -8.4636856577 -8.4622113230 -8.4595009418 -8.4632557058 -8.4606403576 -8.4584318555 -8.457253197 -8.4546085454
SC(n)  -8.4635349966 -8.4590449778 -8.4546297934 -8.4508914927 -8.4459171454 -8.4474079433 -8.4425286291 -8.4380561609 -8.434613536 -8.4297049187
FPE(n)  0.0002095064  0.0002096907  0.0002098595  0.0002098863  0.0002101727  0.0002091032  0.0002093687  0.0002095492  0.000209514  0.0002097862
#AIC lag is 6
selectSETAR(LRHSI, m=2, thDelay = 1)
Using maximum autoregressive order for low regime: mL = 2 
Using maximum autoregressive order for high regime: mH = 2 
Searching on 1031 possible threshold values within regimes with sufficient ( 15% ) number of observations
Searching on  4124  combinations of thresholds ( 1031 ), thDelay ( 1 ), mL ( 2 ) and MM ( 2 ) 
Results of the grid search for 1 threshold

mod.setar4 = setar(LRHSI, m=2, thDelay = 1, th= -0.01201875)
summary(mod.setar4)

Non linear autoregressive model

SETAR model ( 2 regimes)
Coefficients:
Low regime:
    const.L      phiL.1      phiL.2 
-0.01232914  0.03013418 -0.51845312 

High regime:
      const.H        phiH.1        phiH.2 
-6.151951e-05 -1.973568e-02 -2.228439e-02 

Threshold:
-Variable: Z(t) = + (0) X(t)+ (1)X(t-1)
-Value: -0.01202 (fixed)
Proportion of points in low regime: 18.67%   High regime: 81.33% 

Residuals:
        Min          1Q      Median          3Q         Max 
-0.06052237 -0.00828857  0.00055994  0.00774677  0.08113004 

Fit:
residuals variance = 0.0002036,  AIC = -12525, MAPE = 122.8%

Coefficient(s):

           Estimate  Std. Error  t value  Pr(>|t|)    
const.L -0.01232914  0.00224887  -5.4824 4.933e-08 ***
phiL.1   0.03013417  0.05287083   0.5700    0.5688    
phiL.2  -0.51845312  0.10001403  -5.1838 2.477e-07 ***
const.H -0.00006152  0.00044406  -0.1385    0.8898    
phiH.1  -0.01973568  0.02970265  -0.6644    0.5065    
phiH.2  -0.02228439  0.03762699  -0.5922    0.5538    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold
Variable: Z(t) = + (0) X(t) + (1) X(t-1)

Value: -0.01202 (fixed)

Explanation:

AIC order: 10
Hence Threshold value: -0.01201875

Implication: The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series’ behavior.

N100:

VARselect(LRN100)
$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     8      8      1      8 

$criteria
                   1             2            3             4             5             6             7             8             9            10
AIC(n) -8.9021481492 -8.9043488399 -8.903201533 -8.9023525834 -8.9026105334 -8.9126158023 -8.9197348485 -8.9211745124 -8.9199938873 -8.9187080865
HQ(n)  -8.8995478063 -8.9004483257 -8.898000847 -8.8958517263 -8.8948095049 -8.9035146024 -8.9093334771 -8.9094729696 -8.9069921731 -8.9044062009
SC(n)  -8.8951620348 -8.8938696685 -8.889229304 -8.8848872977 -8.8816521905 -8.8881644023 -8.8917903913 -8.8897369980 -8.8850633158 -8.8802844579
FPE(n)  0.0001360963  0.0001357971  0.000135953  0.0001360684  0.0001360333  0.0001346791  0.0001337237  0.0001335313  0.0001336891  0.0001338611
#AIC lag is 8
selectSETAR(LRN100, m=2, thDelay = 1)
Using maximum autoregressive order for low regime: mL = 2 
Using maximum autoregressive order for high regime: mH = 2 
Searching on 1072 possible threshold values within regimes with sufficient ( 15% ) number of observations
Searching on  4288  combinations of thresholds ( 1072 ), thDelay ( 1 ), mL ( 2 ) and MM ( 2 ) 
Results of the grid search for 1 threshold

mod.setar5 = setar(LRN100, m=2, thDelay = 1, th= -0.006764070)
summary(mod.setar5)

Non linear autoregressive model

SETAR model ( 2 regimes)
Coefficients:
Low regime:
    const.L      phiL.1      phiL.2 
0.002177095 0.112955295 0.142519637 

High regime:
      const.H        phiH.1        phiH.2 
-2.901857e-05 -5.224197e-02  7.378584e-02 

Threshold:
-Variable: Z(t) = + (0) X(t)+ (1)X(t-1)
-Value: -0.006764 (fixed)
Proportion of points in low regime: 19.23%   High regime: 80.77% 

Residuals:
        Min          1Q      Median          3Q         Max 
-0.12701096 -0.00470824  0.00063317  0.00590786  0.07355260 

Fit:
residuals variance = 0.0001335,  AIC = -13692, MAPE = 134.3%

Coefficient(s):

           Estimate  Std. Error  t value Pr(>|t|)  
const.L  2.1771e-03  1.1339e-03   1.9200  0.05505 .
phiL.1   1.1296e-01  4.6231e-02   2.4433  0.01467 *
phiL.2   1.4252e-01  5.7836e-02   2.4642  0.01384 *
const.H -2.9019e-05  3.7018e-04  -0.0784  0.93753  
phiH.1  -5.2242e-02  3.0466e-02  -1.7147  0.08659 .
phiH.2   7.3786e-02  4.2470e-02   1.7374  0.08253 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold
Variable: Z(t) = + (0) X(t) + (1) X(t-1)

Value: -0.006764 (fixed)

Explanation:

AIC order: 10
Hence Threshold value: -0.006764070

Implication: The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series’ behavior.

Random Walk and Market Efficiency Check for all 5 returns series:

Test 1: Automatic Variance Ratio Test

H0: follows random walk and has weak form of mkt efficiency i.e. mkt is not predictable

if critical value > 1.96, we reject the null hypothesis

Test 2: Automatic Portmanteau Test

H0: follows random walk and has weak form of mkt efficiency i.e. mkt is not predictable

if critical value > 1.96, we reject the null hypothesis

Test 3: Hurst Exponent Test

H= 0.5 or near : follows random walk

H= 0 or near: short memory

H= 1 or near: long memory

BSESN:

# RANDOM WALK MODEL
Auto.VR(LRBSESN)
$stat
[1] 28.85623

$sum
[1] 2.503848
Auto.Q(LRBSESN)
$Stat
[1] 0.4566666

$Pvalue
[1] 0.499186
hurstexp(LRBSESN)
Simple R/S Hurst estimation:         0.5548493 
Corrected R over S Hurst exponent:   0.5783945 
Empirical Hurst exponent:            0.5033265 
Corrected empirical Hurst exponent:  0.4747791 
Theoretical Hurst exponent:          0.5329845 

Explanation:

Automatic Variance Ratio test suggests series does not follow random walk model

Automatic Portmanteau Test suggests series follows random walk model

Hurst Exponent suggests series follows random walk model

Automatic Variance Ratio test is more reliable hence we say the NSEI returns series does not follow random walk and stock has some degree of market inefficiency

GSPC:

Auto.VR(LRGSPC)
$stat
[1] 46.63477

$sum
[1] 4.061747
Auto.Q(LRGSPC)
$Stat
[1] 4.901365

$Pvalue
[1] 0.02683548
hurstexp(LRGSPC)
Simple R/S Hurst estimation:         0.5215629 
Corrected R over S Hurst exponent:   0.5472432 
Empirical Hurst exponent:            0.5406975 
Corrected empirical Hurst exponent:  0.5025474 
Theoretical Hurst exponent:          0.5369649 

Explanation:

Automatic Variance Ratio test suggests series does not follow random walk model

Automatic Portmanteau Test suggests series does not follow random walk model

Hurst Exponent suggests series follows random walk model

Automatic Variance Ratio test is more reliable hence we say the GSPC returns series does not follow random walk and stock has some degree of market inefficiency

N225:

Auto.VR(LRN225)
$stat
[1] 9.521586

$sum
[1] 1.368356
Auto.Q(LRN225)
$Stat
[1] 0.02998929

$Pvalue
[1] 0.8625145
hurstexp(LRN225)
Simple R/S Hurst estimation:         0.5241724 
Corrected R over S Hurst exponent:   0.5420117 
Empirical Hurst exponent:            0.5083791 
Corrected empirical Hurst exponent:  0.4733481 
Theoretical Hurst exponent:          0.5371811 

Explanation:

Automatic Variance Ratio test suggests series does not follow random walk model

Automatic Portmanteau Test suggests series does not follow random walk model

Hurst Exponent suggests series follows random walk model

Automatic Variance Ratio test is more reliable hence we say the GSPC returns series does not follow random walk and stock has some degree of market inefficiency

HSI:

Auto.VR(LRHSI)
$stat
[1] -0.0555631

$sum
[1] 0.9981716
Auto.Q(LRHSI)
$Stat
[1] 0.01284963

$Pvalue
[1] 0.9097482
hurstexp(LRHSI)
Simple R/S Hurst estimation:         0.5056465 
Corrected R over S Hurst exponent:   0.5226207 
Empirical Hurst exponent:            0.5192194 
Corrected empirical Hurst exponent:  0.4855209 
Theoretical Hurst exponent:          0.5329845 

Explanation:

Automatic Variance Ratio test suggests series follows random walk model

Automatic Portmanteau Test suggests series follows random walk model

Hurst Exponent suggests series follows random walk model

From all three tests we can say the HSI returns series follows random walk and stock has weak form of market efficiency

N100:

Auto.VR(LRN100)
$stat
[1] -0.5221212

$sum
[1] 0.9832896
Auto.Q(LRN100)
$Stat
[1] 0.008651063

$Pvalue
[1] 0.9258948
hurstexp(LRN100)
Simple R/S Hurst estimation:         0.5260294 
Corrected R over S Hurst exponent:   0.5526032 
Empirical Hurst exponent:            0.558224 
Corrected empirical Hurst exponent:  0.5210087 
Theoretical Hurst exponent:          0.5354925 

Explanation:

Automatic Variance Ratio test suggests series follows random walk model

Automatic Portmanteau Test suggests series follows random walk model

Hurst Exponent suggests series follows random walk model

From all the 3 tests we say the N100 returns series follows random walk and stock has weak form of market efficiency

ARCH & GARCH Test for all 5 series:

To include stylised facts of returns series into forecasting, we will move ahead to forecasting with ARCH & GARCH for all of our returns series

ARCH Test:

AutoRegressive Conditional Heteroskedasticity Test

H0: There is no ARCH effect in the returns series of data

H1: There is ARCH effect in the returns series of data

BSESN:

#step 2: ARCH effect test
ArchTest(LRBSESN)  #p-value < 0.05, there is ARCH effect

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LRBSESN
Chi-squared = 463.03, df = 12, p-value < 2.2e-16
#there is ARCH effect

# step3: ARCH/GARCH order
garch(LRBSESN, grad="numerical",trace=FALSE)

Call:
garch(x = LRBSESN, grad = "numerical", trace = FALSE)

Coefficient(s):
       a0         a1         b1  
9.339e-05  5.000e-02  5.000e-02  
#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit=ugarchfit(x,data=LRBSESN)
x_fit

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000877    0.000210   4.1743  0.00003
omega   0.000002    0.000001   1.5114  0.13068
alpha1  0.115395    0.019516   5.9130  0.00000
beta1   0.868095    0.019976  43.4574  0.00000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000877    0.000225  3.90409 0.000095
omega   0.000002    0.000005  0.40682 0.684136
alpha1  0.115395    0.056525  2.04149 0.041202
beta1   0.868095    0.062088 13.98179 0.000000

LogLikelihood : 4821.678 

Information Criteria
------------------------------------
                    
Akaike       -6.5324
Bayes        -6.5181
Shibata      -6.5325
Hannan-Quinn -6.5271

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic  p-value
Lag[1]                      7.278 0.006980
Lag[2*(p+q)+(p+q)-1][2]     7.490 0.008834
Lag[4*(p+q)+(p+q)-1][5]     9.124 0.015397
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1916  0.6616
Lag[2*(p+q)+(p+q)-1][5]    0.4586  0.9639
Lag[4*(p+q)+(p+q)-1][9]    1.6331  0.9440
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1965 0.500 2.000  0.6575
ARCH Lag[5]    0.3953 1.440 1.667  0.9140
ARCH Lag[7]    0.6795 2.315 1.543  0.9594

Nyblom stability test
------------------------------------
Joint Statistic:  28.2795
Individual Statistics:              
mu     0.02716
omega  3.31493
alpha1 0.19591
beta1  0.15197

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     37.31     0.007252
2    30     51.54     0.006128
3    40     55.93     0.038626
4    50     66.19     0.051312


Elapsed time : 0.0618391 

News Impact Curve

x=newsimpact(x_fit)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)

e-GARCH (For asymmetry)

model1=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel1=ugarchfit(model1,data=LRBSESN)
fitmodel1

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : eGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.000532    0.000160    3.3313 0.000864
omega  -0.237171    0.005761  -41.1672 0.000000
alpha1 -0.102849    0.008139  -12.6370 0.000000
beta1   0.974429    0.000798 1221.8433 0.000000
gamma1  0.163178    0.011168   14.6106 0.000000

Robust Standard Errors:
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.000532    0.000144    3.6956 0.000219
omega  -0.237171    0.008609  -27.5496 0.000000
alpha1 -0.102849    0.016763   -6.1355 0.000000
beta1   0.974429    0.000799 1218.9470 0.000000
gamma1  0.163178    0.029997    5.4399 0.000000

LogLikelihood : 4846.039 

Information Criteria
------------------------------------
                    
Akaike       -6.5641
Bayes        -6.5462
Shibata      -6.5641
Hannan-Quinn -6.5574

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic  p-value
Lag[1]                      8.806 0.003002
Lag[2*(p+q)+(p+q)-1][2]     8.944 0.003636
Lag[4*(p+q)+(p+q)-1][5]    10.464 0.006993
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.3962  0.5291
Lag[2*(p+q)+(p+q)-1][5]    0.6467  0.9329
Lag[4*(p+q)+(p+q)-1][9]    1.9671  0.9089
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.06806 0.500 2.000  0.7942
ARCH Lag[5]   0.48080 1.440 1.667  0.8893
ARCH Lag[7]   0.58915 2.315 1.543  0.9696

Nyblom stability test
------------------------------------
Joint Statistic:  1.0483
Individual Statistics:             
mu     0.1828
omega  0.2502
alpha1 0.3763
beta1  0.2403
gamma1 0.1541

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.28 1.47 1.88
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     36.91     0.008155
2    30     40.32     0.078913
3    40     52.13     0.077753
4    50     52.63     0.335517


Elapsed time : 0.06346011 

Explanation:

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is gamma value > 0 i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.

ugarchforecast(x_fit, n.ahead=20)

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 20
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=2023-12-29]:
        Series    Sigma
T+1  0.0008769 0.007402
T+2  0.0008769 0.007489
T+3  0.0008769 0.007574
T+4  0.0008769 0.007656
T+5  0.0008769 0.007737
T+6  0.0008769 0.007815
T+7  0.0008769 0.007891
T+8  0.0008769 0.007965
T+9  0.0008769 0.008037
T+10 0.0008769 0.008107
T+11 0.0008769 0.008176
T+12 0.0008769 0.008243
T+13 0.0008769 0.008308
T+14 0.0008769 0.008372
T+15 0.0008769 0.008434
T+16 0.0008769 0.008495
T+17 0.0008769 0.008554
T+18 0.0008769 0.008612
T+19 0.0008769 0.008669
T+20 0.0008769 0.008724

GSPC:

#step 2: ARCH effect test
ArchTest(LRGSPC)  #p-value < 0.05, there is ARCH effect

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LRGSPC
Chi-squared = 593.04, df = 12, p-value < 2.2e-16
#there is ARCH effect
# step3: ARCH/GARCH order
garch(LRGSPC, grad="numerical",trace=FALSE)

Call:
garch(x = LRGSPC, grad = "numerical", trace = FALSE)

Coefficient(s):
       a0         a1         b1  
0.0001141  0.0500000  0.0500000  
#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit1=ugarchfit(x,data=LRGSPC)
x_fit1

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000913    0.000215   4.2498 0.000021
omega   0.000005    0.000004   1.3576 0.174600
alpha1  0.201158    0.027412   7.3384 0.000000
beta1   0.774701    0.032942  23.5168 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000913    0.000300  3.04710 0.002311
omega   0.000005    0.000024  0.20389 0.838438
alpha1  0.201158    0.088012  2.28557 0.022279
beta1   0.774701    0.166989  4.63924 0.000003

LogLikelihood : 4793.509 

Information Criteria
------------------------------------
                    
Akaike       -6.3521
Bayes        -6.3380
Shibata      -6.3521
Hannan-Quinn -6.3469

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.03961  0.8422
Lag[2*(p+q)+(p+q)-1][2]   0.04905  0.9571
Lag[4*(p+q)+(p+q)-1][5]   0.36408  0.9765
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.04412  0.8336
Lag[2*(p+q)+(p+q)-1][5]   0.84524  0.8936
Lag[4*(p+q)+(p+q)-1][9]   2.00138  0.9048
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.02466 0.500 2.000  0.8752
ARCH Lag[5]   1.69851 1.440 1.667  0.5417
ARCH Lag[7]   2.22367 2.315 1.543  0.6700

Nyblom stability test
------------------------------------
Joint Statistic:  4.0087
Individual Statistics:              
mu     0.06404
omega  0.33970
alpha1 0.06068
beta1  0.16191

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     57.07    1.116e-05
2    30     83.19    3.911e-07
3    40     80.70    9.850e-05
4    50     98.03    4.026e-05


Elapsed time : 0.05446005 

Explanation:

ARCH test suggests that there is ARCH effect in the data hence we can move ahead to do GARCH forecasting of the series

GARCH gave ARCH,GARCH order of 1,1

Sign Bias Test suggest that there is no significant effect of positive and negative bias individually.

but there is collective effect of both the signs on returns of series. hence there is asymmetry in volatility of returns

News Impact Curve

x=newsimpact(x_fit1)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)

e-GARCH (For asymmetry)

model2=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel2=ugarchfit(model2,data=LRGSPC)
fitmodel2

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : eGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000435    0.000219   1.9927 0.046296
omega  -0.495791    0.078389  -6.3247 0.000000
alpha1 -0.150342    0.019505  -7.7078 0.000000
beta1   0.945264    0.008550 110.5608 0.000000
gamma1  0.282000    0.033251   8.4810 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000435    0.000217   2.0049 0.044974
omega  -0.495791    0.107859  -4.5967 0.000004
alpha1 -0.150342    0.039768  -3.7805 0.000157
beta1   0.945264    0.012022  78.6286 0.000000
gamma1  0.282000    0.057996   4.8624 0.000001

LogLikelihood : 4812.87 

Information Criteria
------------------------------------
                    
Akaike       -6.3765
Bayes        -6.3589
Shibata      -6.3765
Hannan-Quinn -6.3699

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.3082  0.5788
Lag[2*(p+q)+(p+q)-1][2]    0.3366  0.7757
Lag[4*(p+q)+(p+q)-1][5]    0.8318  0.8964
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                  4.684e-05  0.9945
Lag[2*(p+q)+(p+q)-1][5] 1.294e+00  0.7902
Lag[4*(p+q)+(p+q)-1][9] 2.727e+00  0.8030
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.02523 0.500 2.000  0.8738
ARCH Lag[5]   2.67608 1.440 1.667  0.3403
ARCH Lag[7]   3.37236 2.315 1.543  0.4461

Nyblom stability test
------------------------------------
Joint Statistic:  1.4007
Individual Statistics:             
mu     0.0446
omega  0.1498
alpha1 0.1765
beta1  0.1425
gamma1 0.4051

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.28 1.47 1.88
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     52.90    4.843e-05
2    30     66.16    9.961e-05
3    40     73.75    6.455e-04
4    50     86.76    7.141e-04


Elapsed time : 0.063097 

Explanation:

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is gamma value > 0 i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.

ugarchforecast(x_fit1, n.ahead=20)

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 20
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=2023-12-29]:
       Series    Sigma
T+1  0.000913 0.006735
T+2  0.000913 0.007014
T+3  0.000913 0.007276
T+4  0.000913 0.007523
T+5  0.000913 0.007756
T+6  0.000913 0.007977
T+7  0.000913 0.008187
T+8  0.000913 0.008386
T+9  0.000913 0.008577
T+10 0.000913 0.008759
T+11 0.000913 0.008932
T+12 0.000913 0.009099
T+13 0.000913 0.009259
T+14 0.000913 0.009412
T+15 0.000913 0.009559
T+16 0.000913 0.009700
T+17 0.000913 0.009836
T+18 0.000913 0.009967
T+19 0.000913 0.010093
T+20 0.000913 0.010215

N225:

#step 2: ARCH effect test
ArchTest(LRN225)  #p-value < 0.05, there is ARCH effect

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LRN225
Chi-squared = 265.76, df = 12, p-value < 2.2e-16
#there is ARCH effect

# step3: ARCH/GARCH order
garch(LRN225, grad="numerical",trace=FALSE)

Call:
garch(x = LRN225, grad = "numerical", trace = FALSE)

Coefficient(s):
       a0         a1         b1  
0.0001238  0.0500000  0.0500000  
#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit2=ugarchfit(x,data=LRN225)
x_fit2

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000521    0.000278   1.8777 0.060427
omega   0.000014    0.000001  23.2136 0.000000
alpha1  0.131004    0.013754   9.5246 0.000000
beta1   0.773102    0.017847  43.3181 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000521    0.000259   2.0124 0.044176
omega   0.000014    0.000001  16.8053 0.000000
alpha1  0.131004    0.015503   8.4504 0.000000
beta1   0.773102    0.021694  35.6368 0.000000

LogLikelihood : 4473.935 

Information Criteria
------------------------------------
                    
Akaike       -6.1106
Bayes        -6.0962
Shibata      -6.1107
Hannan-Quinn -6.1052

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                   0.003439  0.9532
Lag[2*(p+q)+(p+q)-1][2]  1.477468  0.3662
Lag[4*(p+q)+(p+q)-1][5]  3.145312  0.3812
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2593  0.6106
Lag[2*(p+q)+(p+q)-1][5]    2.2939  0.5508
Lag[4*(p+q)+(p+q)-1][9]    4.9180  0.4416
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.3689 0.500 2.000  0.5436
ARCH Lag[5]    2.8382 1.440 1.667  0.3141
ARCH Lag[7]    4.2026 2.315 1.543  0.3183

Nyblom stability test
------------------------------------
Joint Statistic:  56.6899
Individual Statistics:              
mu     0.04467
omega  7.11029
alpha1 0.09084
beta1  0.10619

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     50.92    9.595e-05
2    30     59.25    7.652e-04
3    40     80.70    9.838e-05
4    50     82.35    2.007e-03


Elapsed time : 0.05044484 

Explanation:

ARCH test suggests that there is ARCH effect in the data hence we can move ahead to do GARCH forecasting of the series

GARCH gave ARCH,GARCH order of 1,1

Sign Bias Test suggest that there is no significant effect of positive and negative bias individually.

but there is collective effect of both the signs on returns of series. hence there is asymmetry in volatility of returns

News Impact Curve

x=newsimpact(x_fit2)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)

e-GARCH (For asymmetry)

model3=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel3=ugarchfit(model3,data=LRN225)
fitmodel3

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : eGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.000158    0.000231   0.68401  0.49397
omega  -0.561271    0.071939  -7.80200  0.00000
alpha1 -0.134601    0.014011  -9.60660  0.00000
beta1   0.936967    0.007979 117.43029  0.00000
gamma1  0.174151    0.022325   7.80064  0.00000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000158    0.000218  0.72381 0.469182
omega  -0.561271    0.112044 -5.00938 0.000001
alpha1 -0.134601    0.023651 -5.69102 0.000000
beta1   0.936967    0.012641 74.12130 0.000000
gamma1  0.174151    0.020713  8.40771 0.000000

LogLikelihood : 4501.276 

Information Criteria
------------------------------------
                    
Akaike       -6.1467
Bayes        -6.1286
Shibata      -6.1467
Hannan-Quinn -6.1399

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1251  0.7236
Lag[2*(p+q)+(p+q)-1][2]    2.0599  0.2530
Lag[4*(p+q)+(p+q)-1][5]    3.9440  0.2611
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.9429  0.3315
Lag[2*(p+q)+(p+q)-1][5]    3.3893  0.3405
Lag[4*(p+q)+(p+q)-1][9]    6.0086  0.2972
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1015 0.500 2.000  0.7500
ARCH Lag[5]    2.5692 1.440 1.667  0.3587
ARCH Lag[7]    4.1788 2.315 1.543  0.3215

Nyblom stability test
------------------------------------
Joint Statistic:  1.0639
Individual Statistics:             
mu     0.1472
omega  0.1662
alpha1 0.4928
beta1  0.1686
gamma1 0.1584

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.28 1.47 1.88
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     34.95     0.014164
2    30     45.22     0.027990
3    40     62.17     0.010593
4    50     76.27     0.007574


Elapsed time : 0.08286595 

Explanation:

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is gamma value > 0 i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.

ugarchforecast(x_fit2, n.ahead=20)

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 20
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=2023-12-29]:
        Series    Sigma
T+1  0.0005211 0.009611
T+2  0.0005211 0.009860
T+3  0.0005211 0.010080
T+4  0.0005211 0.010274
T+5  0.0005211 0.010447
T+6  0.0005211 0.010601
T+7  0.0005211 0.010738
T+8  0.0005211 0.010860
T+9  0.0005211 0.010970
T+10 0.0005211 0.011068
T+11 0.0005211 0.011156
T+12 0.0005211 0.011235
T+13 0.0005211 0.011306
T+14 0.0005211 0.011369
T+15 0.0005211 0.011427
T+16 0.0005211 0.011478
T+17 0.0005211 0.011525
T+18 0.0005211 0.011566
T+19 0.0005211 0.011604
T+20 0.0005211 0.011638

HSI:

#step 2: ARCH effect test
ArchTest(LRHSI)  #p-value < 0.05, there is ARCH effect

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LRHSI
Chi-squared = 157.75, df = 12, p-value < 2.2e-16
#there is ARCH effect

# step3: ARCH/GARCH order
garch(LRHSI, grad="numerical",trace=FALSE)

Call:
garch(x = LRHSI, grad = "numerical", trace = FALSE)

Coefficient(s):
       a0         a1         b1  
0.0001772  0.0500000  0.0500000  
#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit3=ugarchfit(x,data=LRHSI)
x_fit3

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000196    0.000334  -0.5869 0.557273
omega   0.000005    0.000001   4.0367 0.000054
alpha1  0.062641    0.006623   9.4586 0.000000
beta1   0.910551    0.009264  98.2937 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000196    0.000334 -0.58779 0.556676
omega   0.000005    0.000003  2.11236 0.034656
alpha1  0.062641    0.009853  6.35735 0.000000
beta1   0.910551    0.013349 68.21104 0.000000

LogLikelihood : 4243.524 

Information Criteria
------------------------------------
                    
Akaike       -5.7485
Bayes        -5.7341
Shibata      -5.7485
Hannan-Quinn -5.7432

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.02601  0.8719
Lag[2*(p+q)+(p+q)-1][2]   0.07051  0.9408
Lag[4*(p+q)+(p+q)-1][5]   0.73939  0.9152
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.09483  0.7581
Lag[2*(p+q)+(p+q)-1][5]   0.79490  0.9041
Lag[4*(p+q)+(p+q)-1][9]   2.66702  0.8124
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1784 0.500 2.000  0.6727
ARCH Lag[5]    0.2652 1.440 1.667  0.9494
ARCH Lag[7]    2.3953 2.315 1.543  0.6341

Nyblom stability test
------------------------------------
Joint Statistic:  4.4769
Individual Statistics:              
mu     0.03458
omega  0.15818
alpha1 0.37731
beta1  0.30034

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     44.61    0.0007778
2    30     52.32    0.0050263
3    40     57.39    0.0289508
4    50     75.14    0.0095687


Elapsed time : 0.04693198 

Explanation:

ARCH test suggests that there is ARCH effect in the data hence we can move ahead to do GARCH forecasting of the series

GARCH gave ARCH,GARCH order of 1,1

The lack of significant overall sign bias and individual sign biases suggests that the impact of positive and negative shocks on the variance is symmetric. This means that positive and negative shocks have similar effects on the volatility of the time series.

News Impact Curve

x=newsimpact(x_fit3)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)

e-GARCH (For asymmetricity)

model4=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel4=ugarchfit(model3,data=LRHSI)
fitmodel4

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : eGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu     -0.000512    0.000307   -1.6659 0.095743
omega  -0.253424    0.003643  -69.5739 0.000000
alpha1 -0.077708    0.010569   -7.3526 0.000000
beta1   0.970123    0.000655 1481.4063 0.000000
gamma1  0.109938    0.007070   15.5491 0.000000

Robust Standard Errors:
        Estimate  Std. Error   t value Pr(>|t|)
mu     -0.000512    0.000286   -1.7881 0.073752
omega  -0.253424    0.008900  -28.4751 0.000000
alpha1 -0.077708    0.016703   -4.6523 0.000003
beta1   0.970123    0.000889 1090.9301 0.000000
gamma1  0.109938    0.016011    6.8662 0.000000

LogLikelihood : 4257.981 

Information Criteria
------------------------------------
                    
Akaike       -5.7668
Bayes        -5.7488
Shibata      -5.7668
Hannan-Quinn -5.7601

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.03958  0.8423
Lag[2*(p+q)+(p+q)-1][2]   0.11438  0.9097
Lag[4*(p+q)+(p+q)-1][5]   0.76697  0.9097
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.7742  0.3789
Lag[2*(p+q)+(p+q)-1][5]    1.9970  0.6191
Lag[4*(p+q)+(p+q)-1][9]    3.4799  0.6778
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.5112 0.500 2.000  0.4746
ARCH Lag[5]    0.6486 1.440 1.667  0.8391
ARCH Lag[7]    1.9804 2.315 1.543  0.7215

Nyblom stability test
------------------------------------
Joint Statistic:  0.8802
Individual Statistics:              
mu     0.06105
omega  0.27372
alpha1 0.10943
beta1  0.25554
gamma1 0.06889

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.28 1.47 1.88
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     39.59     0.003703
2    30     44.55     0.032553
3    40     51.43     0.087835
4    50     68.02     0.037338


Elapsed time : 0.06761289 

Explanation:

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is gamma value > 0 i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.

ugarchforecast(x_fit3, n.ahead=20)

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 20
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=2023-12-29]:
         Series   Sigma
T+1  -0.0001961 0.01394
T+2  -0.0001961 0.01395
T+3  -0.0001961 0.01396
T+4  -0.0001961 0.01397
T+5  -0.0001961 0.01398
T+6  -0.0001961 0.01399
T+7  -0.0001961 0.01400
T+8  -0.0001961 0.01401
T+9  -0.0001961 0.01402
T+10 -0.0001961 0.01402
T+11 -0.0001961 0.01403
T+12 -0.0001961 0.01404
T+13 -0.0001961 0.01405
T+14 -0.0001961 0.01405
T+15 -0.0001961 0.01406
T+16 -0.0001961 0.01407
T+17 -0.0001961 0.01407
T+18 -0.0001961 0.01408
T+19 -0.0001961 0.01409
T+20 -0.0001961 0.01409

HSI:

#step 2: ARCH effect test
ArchTest(LRN100)  #p-value < 0.05, there is ARCH effect

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LRN100
Chi-squared = 372.17, df = 12, p-value < 2.2e-16
#there is ARCH effect

# step3: ARCH/GARCH order
garch(LRN100, grad="numerical",trace=FALSE)

Call:
garch(x = LRN100, grad = "numerical", trace = FALSE)

Coefficient(s):
       a0         a1         b1  
0.0001117  0.0500000  0.0500000  
#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit4=ugarchfit(x,data=LRN100)
x_fit4

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000679    0.000217   3.1290 0.001754
omega   0.000008    0.000001   9.7011 0.000000
alpha1  0.209428    0.024275   8.6274 0.000000
beta1   0.737770    0.020734  35.5828 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000679    0.000230   2.9476 0.003203
omega   0.000008    0.000001   6.4120 0.000000
alpha1  0.209428    0.027947   7.4938 0.000000
beta1   0.737770    0.033257  22.1841 0.000000

LogLikelihood : 4943.166 

Information Criteria
------------------------------------
                    
Akaike       -6.4312
Bayes        -6.4173
Shibata      -6.4312
Hannan-Quinn -6.4260

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.9015  0.3424
Lag[2*(p+q)+(p+q)-1][2]    0.9038  0.5305
Lag[4*(p+q)+(p+q)-1][5]    1.2244  0.8072
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.5102  0.4750
Lag[2*(p+q)+(p+q)-1][5]    3.7320  0.2894
Lag[4*(p+q)+(p+q)-1][9]    5.3838  0.3753
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]     2.633 0.500 2.000  0.1047
ARCH Lag[5]     3.843 1.440 1.667  0.1887
ARCH Lag[7]     4.432 2.315 1.543  0.2886

Nyblom stability test
------------------------------------
Joint Statistic:  12.2069
Individual Statistics:              
mu     0.02216
omega  3.51522
alpha1 0.16553
beta1  0.27751

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     96.47    2.318e-12
2    30    117.01    1.573e-12
3    40    123.69    9.729e-11
4    50    155.02    6.189e-13


Elapsed time : 0.04173994 

News Impact Curve

x=newsimpact(x_fit4)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)

e-GARCH (For asymmetricity)

model5=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel5=ugarchfit(model5,data=LRN100)
fitmodel5

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : eGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.000064    0.000206   0.31051  0.75617
omega  -0.490254    0.010383 -47.21652  0.00000
alpha1 -0.187089    0.012848 -14.56191  0.00000
beta1   0.946416    0.001416 668.57653  0.00000
gamma1  0.188822    0.009766  19.33541  0.00000

Robust Standard Errors:
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.000064    0.000190   0.33759  0.73567
omega  -0.490254    0.027546 -17.79752  0.00000
alpha1 -0.187089    0.020168  -9.27642  0.00000
beta1   0.946416    0.002736 345.90298  0.00000
gamma1  0.188822    0.029692   6.35926  0.00000

LogLikelihood : 4981.4 

Information Criteria
------------------------------------
                    
Akaike       -6.4797
Bayes        -6.4623
Shibata      -6.4797
Hannan-Quinn -6.4732

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.8021  0.3705
Lag[2*(p+q)+(p+q)-1][2]    0.8060  0.5657
Lag[4*(p+q)+(p+q)-1][5]    1.0173  0.8558
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                      1.689  0.1938
Lag[2*(p+q)+(p+q)-1][5]     3.781  0.2827
Lag[4*(p+q)+(p+q)-1][9]     5.380  0.3758
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]     2.327 0.500 2.000  0.1272
ARCH Lag[5]     3.943 1.440 1.667  0.1791
ARCH Lag[7]     4.559 2.315 1.543  0.2731

Nyblom stability test
------------------------------------
Joint Statistic:  1.0781
Individual Statistics:              
mu     0.27220
omega  0.32521
alpha1 0.05221
beta1  0.30107
gamma1 0.36109

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.28 1.47 1.88
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     74.44    1.654e-08
2    30     87.28    9.485e-08
3    40    104.10    7.754e-08
4    50    120.51    5.872e-08


Elapsed time : 0.06582999 

Explanation:

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is gamma value > 0 i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.

ugarchforecast(x_fit4, n.ahead=20)

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 20
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=2023-12-29]:
        Series    Sigma
T+1  0.0006786 0.005967
T+2  0.0006786 0.006434
T+3  0.0006786 0.006847
T+4  0.0006786 0.007216
T+5  0.0006786 0.007550
T+6  0.0006786 0.007852
T+7  0.0006786 0.008129
T+8  0.0006786 0.008382
T+9  0.0006786 0.008615
T+10 0.0006786 0.008830
T+11 0.0006786 0.009029
T+12 0.0006786 0.009214
T+13 0.0006786 0.009385
T+14 0.0006786 0.009545
T+15 0.0006786 0.009694
T+16 0.0006786 0.009833
T+17 0.0006786 0.009962
T+18 0.0006786 0.010084
T+19 0.0006786 0.010197
T+20 0.0006786 0.010304
---
title: "Financial Econometrics Project - Forecasting 5 Equity Stock Exchanges Globally
  with reports on each."
output:
  html_notebook: default
  pdf_document: default
---

[**Installing and Loading Required Libraries**]{.underline}

```{r}
# # Required Packages
packages = c('quantmod','car','forecast','tseries','FinTS', 'rugarch','utf8','ggplot2')
# 
# # Install all Packages with Dependencies
# install.packages(packages, dependencies = TRUE)
# 
# # Load all Packages
lapply(packages, require, character.only = TRUE)
library(writexl)
library(tseries)
library(TSstudio)
library(fBasics)
library(rcompanion)
library(forecast)
library(lmtest)
library(tsDyn)
library(vars)
library(PerformanceAnalytics)
library(vrtest)
library(pracma)
library(rugarch)
library(FinTS)
library(e1071)
library(readxl)
```

```{r}
# Fetching Data
symbols <- c("^BSESN","^GSPC","^N225","^HSI","^N100")
getSymbols(Symbols = symbols,
           src = 'yahoo',
           from = as.Date('2018-01-01'),
           to = as.Date('2023-12-31'),
           periodicity = 'daily')
```

The names of the stock exchanges selected

	1.	**\^BSESN** - SENSEX (BSE Sensex) of the **Bombay Stock Exchange (BSE)-**

The Bombay Stock Exchange (BSE), located in Mumbai, India, is one of the oldest stock exchanges in Asia, founded in 1875. The BSE Sensex, or SENSEX, is its benchmark index, representing 30 of the largest and most actively traded stocks on the exchange, reflecting the overall performance of the Indian stock market.

	2.	**\^GSPC** - S&P 500 Index of the **New York Stock Exchange (NYSE) and NASDAQ**

The S&P 500 Index tracks the performance of 500 large companies listed on the New York Stock Exchange (NYSE) and NASDAQ. It is widely regarded as one of the best gauges of the U.S. stock market’s health and is a common benchmark for investment performance.

	3.	**\^N225** - Nikkei 225 Index of the **Tokyo Stock Exchange**

The Nikkei 225 is a stock market index for the Tokyo Stock Exchange (TSE), Japan’s premier stock exchange. It comprises 225 of the largest and most liquid stocks in Japan, serving as a key indicator of the Japanese economy and stock market trends.

	4.	**\^HSI** - Hang Seng Index of the **Hong Kong Stock Exchange (HKEX)**

The Hang Seng Index (HSI) is the main stock market index of the Hong Kong Stock Exchange (HKEX). It tracks the performance of the largest and most liquid companies listed in Hong Kong, providing insight into the broader economic health of Hong Kong and China.

	5.	**\^N100** - Euronext 100 Index of the **Euronext Stock Exchange** (covers major European markets)

The Euronext 100 Index represents the largest and most liquid stocks traded on the Euronext Stock Exchange, which operates in several European countries, including France, Belgium, Netherlands, and Portugal. The index includes 100 blue-chip companies, reflecting the performance of major European markets.

**Cleaning the Data**

```{r}
# Cleaning Data
BSESN <- na.omit(BSESN$BSESN.Close)
GSPC <- na.omit(GSPC$GSPC.Close)
N225 <- na.omit(N225$N225.Close)
HSI <- na.omit(HSI$HSI.Close)
N100 <- na.omit(N100$N100.Close)
```

**Taking Log Returns of the Chosen Exchanges and Testing Stationarity**

Checking Stationarity of the Data:

Used Test: **Augmented Dicky Fuller's Test**

**H0**: The series is not stationary

**H1**: The series is stationary

1.  **BSESN**

```{r}
# Log Differencing and ADF Test
LRBSESN <- diff(log(BSESN))
LRBSESN <- na.omit(LRBSESN)
adf.test(LRBSESN)
```

**Results -** p-value \<0.05 Hence We Reject H0

2.  **GSPC**

```{r}
LRGSPC <- diff(log(GSPC))
LRGSPC <- na.omit(LRGSPC)
adf.test(LRGSPC)
```

**Results -** p-value \<0.05 Hence We Reject H0

3.  **N225**

```{r}
LRN225 <- diff(log(N225))
LRN225 <- na.omit(LRN225)
adf.test(LRN225)
```

**Results -** p-value \<0.05 Hence We Reject H0

4.  **HSI**

```{r}
LRHSI <- diff(log(HSI))
LRHSI <- na.omit(LRHSI)
adf.test(LRHSI)
```

**Results -** p-value \<0.05 Hence We Reject H0

5.  **N100**

```{r}
LRN100 <- diff(log(N100))
LRN100 <- na.omit(LRN100)
adf.test(LRN100)
```

**Results -** p-value \<0.05 Hence We Reject H0

**Normality Test** for all Series:

Used Test: **Jarque Bera Test**

**H0**: The series is normally ditsributed

**H1**: The series is not normally distributed

```{r}
jarque.bera.test(LRBSESN)
jarque.bera.test(LRGSPC)
jarque.bera.test(LRN225)
jarque.bera.test(LRHSI)
jarque.bera.test(LRN100)
```

**Explanation:**

p-value for all 5 series is less than 0.05

hence we reject the null hypothesis and say that all the series are not normally distributed

Implication: all the 5 return series are risky stock exchanges

**Plotting all 5 Stock Exchanges**

```{r}
plotNormalHistogram(LRBSESN)
plotNormalDensity(LRBSESN)
plotNormalHistogram(LRGSPC)
plotNormalDensity(LRGSPC)
plotNormalHistogram(LRN225)
plotNormalDensity(LRN225)
plotNormalHistogram(LRHSI)
plotNormalDensity(LRHSI)
plotNormalHistogram(LRN100)
plotNormalDensity(LRN100)
```

**Obtaining Basic Statistics of Log Returns of Each Stock Exchanges**

```{r}
# Print Basic Stats 
basicStats(LRBSESN)
basicStats(LRGSPC)
basicStats(LRN225)
basicStats(LRHSI)
basicStats(LRN100)
```

**ARIMA Modelling of Return Series**

1.**BSESN**

```{r}
acf(LRBSESN) 
pacf(LRBSESN) 

#Using Auto arima 
arma_pq_LRBSESN = auto.arima(LRBSESN); arma_pq_LRBSESN

#Application of ARIMA

model2 = arima(LRBSESN,order = c(0,0,1))
model2

et = residuals(model2)
et

#Checking of significance of ARMA
coeftest(model2)

#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))
```
**Explanation:**

1.  **Auto ARIMA:** gave order of ARIMA (0,0,1)

    i.e. no autoregressive (AR) terms, no differencing (D) terms, and one moving average (MA) term.

2.  **Log likelihood:** 4468.7 This is a measure of the model's goodness of fit. A higher log likelihood indicates a better fit.

    **AIC:** -8931.39 The Akaike Information Criterion (AIC) is a measure of model complexity and fit. A lower AIC indicates a better model.

    **sigma\^2:** 0.000137 This is the estimated variance of the residuals, representing the model's ability to fit the data. A smaller value suggests a better fit.

    **Implication:** The ARIMA(0,0,1) model provides a parsimonious representation of the LRNSEI time series. The model suggests that the current value of the series is influenced by the previous period's error.

3.  **The significance test** indicate that the MA1 coefficient in the ARIMA(0,0,1) model is not statistically significant. The intercept coefficient is close to being statistically significant, but more evidence is needed to confirm its significance.

**2.GSPC**

```{r}
acf(LRGSPC) 
pacf(LRGSPC) 

#Using Auto arima 
arma_pq_LRGSPC = auto.arima(LRGSPC); arma_pq_LRGSPC

#Application of ARIMA

model2 = arima(LRGSPC,order = c(4,0,4))
model2

et = residuals(model2)
et

#Checking of significance of ARMA
coeftest(model2)

#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))
```
**Explanation:**

1.  **Auto ARIMA:** gave order of ARIMA (4,0,4)

    i.e. no autoregressive (AR) terms, no differencing (D) terms, and one moving average (MA) term.

2.  **Log likelihood:** 4476.69 This is a measure of the model's goodness of fit. A higher log likelihood indicates a better fit.

    **AIC:** -8935.38 The Akaike Information Criterion (AIC) is a measure of model complexity and fit. A lower AIC indicates a better model.

    **sigma\^2:** 0.0001553 This is the estimated variance of the residuals, representing the model's ability to fit the data. A smaller value suggests a better fit.

    **Implication:** The ARIMA(4,0,4) model provides a parsimonious representation of the LRNSEI time series. The model suggests that the current value of the series is influenced by the previous period's error.

3.  **The significance test** indicate that the MA1 coefficient in the ARIMA(4,0,4) model is not statistically significant. The intercept coefficient is close to being statistically significant, but more evidence is needed to confirm its significance.

**3.N225**

```{r}
acf(LRN225) 
pacf(LRN225) 

#Using Auto arima 
arma_pq_LRN225 = auto.arima(LRN225); arma_pq_LRN225

#Application of ARIMA

model2 = arima(LRN225,order = c(0,0,0))
model2

et = residuals(model2)
et

#Checking of significance of ARMA
coeftest(model2)

#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))
```
**Explanation:**

1.  **Auto ARIMA:** gave order of ARIMA (0,0,0)

    i.e. no autoregressive (AR) terms, no differencing (D) terms, and one moving average (MA) term.

2.  **Log likelihood:** 4371.79 This is a measure of the model's goodness of fit. A higher log likelihood indicates a better fit.

    **AIC:** -8741.58 The Akaike Information Criterion (AIC) is a measure of model complexity and fit. A lower AIC indicates a better model.

    **sigma\^2:** 0.0001486 This is the estimated variance of the residuals, representing the model's ability to fit the data. A smaller value suggests a better fit.

    **Implication:** The ARIMA(0,0,0) model provides a parsimonious representation of the LRNSEI time series. The model suggests that the current value of the series is influenced by the previous period's error.

3.  **The significance test** indicate that the MA1 coefficient in the ARIMA(0,0,0) model is not statistically significant. The intercept coefficient is close to being statistically significant, but more evidence is needed to confirm its significance.
**4.HSI**

```{r}
acf(LRHSI) 
pacf(LRHSI) 

#Using Auto arima 
arma_pq_LRHSI = auto.arima(LRHSI); arma_pq_LRHSI

#Application of ARIMA

model2 = arima(LRHSI,order = c(0,0,0))
model2

et = residuals(model2)
et

#Checking of significance of ARMA
coeftest(model2)

#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))
```
**Explanation:**

1.  **Auto ARIMA:** gave order of ARIMA (0,0,0)

    i.e. no autoregressive (AR) terms, no differencing (D) terms, and no moving average (MA) terms.

2.  **sigma\^2**: 0.0002079. This is the estimated variance of the residuals, representing the model's ability to fit the data.

    **Log likelihood:** 4160.01. A higher log likelihood indicates a better fit.

    **AIC:** -8316.03. A lower AIC indicates a better model.

    **Implication:** The ARIMA(0,0,0) model suggests that the LRHSI time series is a white noise process. This indicates that there are no predictable patterns or trends in the data.

3.  **The significance test** The significance test suggests that the intercept coefficient in the ARIMA(0,0,0) model is not statistically significant.
**5.N100**

```{r}
acf(LRN100) 
pacf(LRN100) 

#Using Auto arima 
arma_pq_LRN100 = auto.arima(LRN100); arma_pq_LRN100

#Application of ARIMA

model2 = arima(LRHSI,order = c(0,0,0))
model2

et = residuals(model2)
et

#Checking of significance of ARMA
coeftest(model2)

#point forecast is called out of sample forecast
autoplot(forecast(model2, h=500))
```
**Explanation:**

1.  **Auto ARIMA:** gave order of ARIMA (0,0,0)

    no autoregressive (AR) terms, no differencing (D) terms, and no moving average (MA) terms.

2.  **sigma\^2:** 0.0001351. This is the estimated variance of the residuals, representing the model's ability to fit the data.

    **Log likelihood:** 4662.94. A higher log likelihood indicates a better fit.

    **AIC:** -9321.89. A lower AIC indicates a better model.

    **Implication:** The ARIMA(0,0,0) model suggests that the LRN100 time series is a white noise process. This indicates that there are no predictable patterns or trends in the data.

3.  **The significance test** The significance test suggests that the intercept coefficient in the ARIMA(0,0,0) model is not statistically significant. This supports the conclusion that the LRN100 series is a white noise process with a mean close to zero.



**Linearity Test:**

**Used Test:** BDS Test

**H0:** There is linearity in log returns of series

**H1:** There is no linearity in log returns of series
```{r}
bds_result_BSESN <- bds.test(LRBSESN)

# Print the result
print(bds_result_BSESN)
#BSESN data is non linear
```

```{r}
bds_result_GSPC <- bds.test(LRGSPC)

# Print the result
print(bds_result_GSPC)
#GSPC data is non linear
```

```{r}
bds_result_N225 <- bds.test(LRN225)

# Print the result
print(bds_result_N225)
#N225 data is non linear
```

```{r}
bds_result_HSI <- bds.test(LRHSI)

# Print the result
print(bds_result_HSI)
#HSI data is non linear
```

```{r}
bds_result_N100 <- bds.test(LRN100)

# Print the result
print(bds_result_N100)
#N100 data is non linear
```
**Explanation:**

p-value for all 5 series is less than 0.05

hence we reject the null hypothesis and say that there is no linearity in log returns of any series

Implication: Since series is not linear, a SETAR model might be a more appropriate choice for modeling the log return series. The SETAR model can capture the non-linear relationships and threshold effects present in the data.



**BSESN:**
```{r}
VARselect(LRBSESN)
#AIC lag is 10
selectSETAR(LRBSESN, m=2, thDelay = 1)
mod.setar1 = setar(LRBSESN, m=2, thDelay = 1, th= -0.007846886)
summary(mod.setar1)
```
**Explanation:**

**AIC order:** 10\
**Hence Threshold value:** -0.007846886

**Implication:** The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series' behavior.

**GSPC:**
```{r}
VARselect(LRGSPC)
#AIC lag is 9
selectSETAR(LRGSPC, m=2, thDelay = 1)
mod.setar2 = setar(LRGSPC, m=2, thDelay = 1, th= -0.007264115)
summary(mod.setar2)
```
**Explanation:**

**AIC order:** 9\
**Hence Threshold value:** -0.007264115

**Implication:** The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series' behavior.

**N225:**
```{r}
VARselect(LRN225)
#AIC lag is 7
selectSETAR(LRN225, m=2, thDelay = 1)
mod.setar3 = setar(LRN225, m=2, thDelay = 1, th= -0.008663830)
summary(mod.setar3)
```
**Explanation:**

**AIC order:** 7\
**Hence Threshold value:** -0.008663830

**Implication:** The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series' behavior.


**HSI:**
```{r}
VARselect(LRHSI)
#AIC lag is 6
selectSETAR(LRHSI, m=2, thDelay = 1)
mod.setar4 = setar(LRHSI, m=2, thDelay = 1, th= -0.01201875)
summary(mod.setar4)
```
**Explanation:**

**AIC order:** 10\
**Hence Threshold value:** -0.01201875

**Implication:** The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series' behavior.

**N100:**
```{r}
VARselect(LRN100)
#AIC lag is 8
selectSETAR(LRN100, m=2, thDelay = 1)
mod.setar5 = setar(LRN100, m=2, thDelay = 1, th= -0.006764070)
summary(mod.setar5)
```
**Explanation:**

**AIC order:** 10\
**Hence Threshold value:** -0.006764070

**Implication:** The SETAR model provides a suitable framework for capturing the non-linear dynamics in the time series data. The results indicate significant differences in the dynamics between the high and low regimes, suggesting that the threshold is an important factor in understanding the series' behavior.




**Random Walk and Market Efficiency Check for all 5 returns series:**

**Test 1: Automatic Variance Ratio Test**

H0: follows random walk and has weak form of mkt efficiency i.e. mkt is not predictable

if critical value \> 1.96, we reject the null hypothesis

**Test 2: Automatic Portmanteau Test**

H0: follows random walk and has weak form of mkt efficiency i.e. mkt is not predictable

if critical value \> 1.96, we reject the null hypothesis

**Test 3: Hurst Exponent Test**

H= 0.5 or near : follows random walk

H= 0 or near: short memory

H= 1 or near: long memory


**BSESN:**
```{r}
# RANDOM WALK MODEL
Auto.VR(LRBSESN)
Auto.Q(LRBSESN)
hurstexp(LRBSESN)
```
**Explanation:**

Automatic Variance Ratio test suggests series does not follow random walk model

Automatic Portmanteau Test suggests series follows random walk model

Hurst Exponent suggests series follows random walk model

Automatic Variance Ratio test is more reliable hence we say the NSEI returns series **does not follow random walk and stock has some degree of market inefficiency**


**GSPC:**
```{r}
Auto.VR(LRGSPC)
Auto.Q(LRGSPC)
hurstexp(LRGSPC)
```

**Explanation:**

Automatic Variance Ratio test suggests series does not follow random walk model

Automatic Portmanteau Test suggests series does not follow random walk model

Hurst Exponent suggests series follows random walk model

Automatic Variance Ratio test is more reliable hence we say the GSPC returns series **does not follow random walk and stock has some degree of market inefficiency**

**N225:**
```{r}
Auto.VR(LRN225)
Auto.Q(LRN225)
hurstexp(LRN225)
```

**Explanation:**

Automatic Variance Ratio test suggests series does not follow random walk model

Automatic Portmanteau Test suggests series does not follow random walk model

Hurst Exponent suggests series follows random walk model

Automatic Variance Ratio test is more reliable hence we say the GSPC returns series **does not follow random walk and stock has some degree of market inefficiency**

**HSI:**
```{r}
Auto.VR(LRHSI)
Auto.Q(LRHSI)
hurstexp(LRHSI)
```

**Explanation:**

Automatic Variance Ratio test suggests series follows random walk model

Automatic Portmanteau Test suggests series follows random walk model

Hurst Exponent suggests series follows random walk model

From all three tests we can say the HSI returns series **follows random walk and stock has weak form of market efficiency**

**N100:**
```{r}
Auto.VR(LRN100)
Auto.Q(LRN100)
hurstexp(LRN100)
```
**Explanation:**

Automatic Variance Ratio test suggests series follows random walk model

Automatic Portmanteau Test suggests series follows random walk model

Hurst Exponent suggests series follows random walk model

From all the 3 tests we say the N100 returns series **follows random walk and stock has weak form of market efficiency**



**ARCH & GARCH Test for all 5 series:**

To include stylised facts of returns series into forecasting, we will move ahead to forecasting with ARCH & GARCH for all of our returns series

**ARCH Test:**

AutoRegressive Conditional Heteroskedasticity Test

H0: There is no ARCH effect in the returns series of data

H1: There is ARCH effect in the returns series of data


**BSESN:**
```{r}
#step 2: ARCH effect test
ArchTest(LRBSESN)  #p-value < 0.05, there is ARCH effect
#there is ARCH effect

# step3: ARCH/GARCH order
garch(LRBSESN, grad="numerical",trace=FALSE)

#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit=ugarchfit(x,data=LRBSESN)
x_fit
```
## News Impact Curve
```{r}
x=newsimpact(x_fit)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)
```
## e-GARCH (For asymmetry)
```{r}
model1=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel1=ugarchfit(model1,data=LRBSESN)
fitmodel1
```

**Explanation:**

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is **gamma value \> 0** i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.

```{r}
ugarchforecast(x_fit, n.ahead=20)
```


**GSPC:**
```{r}
#step 2: ARCH effect test
ArchTest(LRGSPC)  #p-value < 0.05, there is ARCH effect
#there is ARCH effect
# step3: ARCH/GARCH order
garch(LRGSPC, grad="numerical",trace=FALSE)

#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit1=ugarchfit(x,data=LRGSPC)
x_fit1
```

**Explanation:**

ARCH test suggests that there is ARCH effect in the data hence we can move ahead to do GARCH forecasting of the series

GARCH gave ARCH,GARCH order of 1,1

Sign Bias Test suggest that there is no significant effect of positive and negative bias individually.

but there is collective effect of both the signs on returns of series. hence there is **asymmetry in volatility of returns**

## News Impact Curve
```{r}
x=newsimpact(x_fit1)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)
```

## e-GARCH (For asymmetry)
```{r}
model2=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel2=ugarchfit(model2,data=LRGSPC)
fitmodel2
```

**Explanation:**

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is **gamma value \> 0** i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.
```{r}
ugarchforecast(x_fit1, n.ahead=20)
```

**N225:**
```{r}
#step 2: ARCH effect test
ArchTest(LRN225)  #p-value < 0.05, there is ARCH effect
#there is ARCH effect

# step3: ARCH/GARCH order
garch(LRN225, grad="numerical",trace=FALSE)
#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit2=ugarchfit(x,data=LRN225)
x_fit2
```
**Explanation:**

ARCH test suggests that there is ARCH effect in the data hence we can move ahead to do GARCH forecasting of the series

GARCH gave ARCH,GARCH order of 1,1

Sign Bias Test suggest that there is no significant effect of positive and negative bias individually.

but there is collective effect of both the signs on returns of series. hence there is **asymmetry in volatility of returns**

## News Impact Curve
```{r}
x=newsimpact(x_fit2)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)
```
## e-GARCH (For asymmetry)
```{r}
model3=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel3=ugarchfit(model3,data=LRN225)
fitmodel3
```
**Explanation:**

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is **gamma value \> 0** i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.
```{r}
ugarchforecast(x_fit2, n.ahead=20)
```

**HSI:**
```{r}
#step 2: ARCH effect test
ArchTest(LRHSI)  #p-value < 0.05, there is ARCH effect
#there is ARCH effect

# step3: ARCH/GARCH order
garch(LRHSI, grad="numerical",trace=FALSE)

#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit3=ugarchfit(x,data=LRHSI)
x_fit3
```
**Explanation:**

ARCH test suggests that there is ARCH effect in the data hence we can move ahead to do GARCH forecasting of the series

GARCH gave ARCH,GARCH order of 1,1

The lack of significant overall sign bias and individual sign biases suggests that the **impact of positive and negative shocks on the variance is symmetric**. This means that positive and negative shocks have similar effects on the volatility of the time series.

## News Impact Curve
```{r}
x=newsimpact(x_fit3)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)
```
## e-GARCH (For asymmetricity)
```{r}
model4=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel4=ugarchfit(model3,data=LRHSI)
fitmodel4
```
**Explanation:**

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is **gamma value \> 0** i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.
```{r}
ugarchforecast(x_fit3, n.ahead=20)
```

**HSI:**
```{r}
#step 2: ARCH effect test
ArchTest(LRN100)  #p-value < 0.05, there is ARCH effect
#there is ARCH effect

# step3: ARCH/GARCH order
garch(LRN100, grad="numerical",trace=FALSE)

#step 4: Application of ARCH and GARCH
x=ugarchspec(variance.model = list(garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
x_fit4=ugarchfit(x,data=LRN100)
x_fit4
```
## News Impact Curve

```{r}
x=newsimpact(x_fit4)
plot(x$zx, x$zy, type="l", lwd=2, col="blue", main="GARCH(1,1) - News Impact", ylab=x$yexpr, xlab=x$xexpr)
```
## e-GARCH (For asymmetricity)
```{r}
model5=ugarchspec(variance.model = list(model="eGARCH",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0)))
fitmodel5=ugarchfit(model5,data=LRN100)
fitmodel5
```

**Explanation:**

e-GARCH test is done when there is asymmetry in volatility of returns and when there is leverage effect

here is **gamma value \> 0** i.e. positive hence there is no leverage effect in the data

hence we cannot forecast with e-GARCH model and hence we will forecast the volatility with GARCH model.
```{r}
ugarchforecast(x_fit4, n.ahead=20)
```
